B. H. Bl-ACKWKI.l.. Bookseller, 50 & 5t, Broad St., Oxford. Digitized by the Internet Archive in 2007 with funding from IVIicrosoft Corporation http://www.archive.org/details/algebraofinvariaOOgraciala THE ALGEBRA OF INVARIANTS aonUon: C. J. CLAY and SONS, CAMBRIDGE UNIVERSITY PRESS WAREHOUSE, AVE MARIA LANE. ffilnsaofa: BO, WELLINGTON STREET. Ifipjiji: F. A. BBOCKHAUS. i^ffa gork: THE MACMILLAN COMPANY. ISombne nnU CTalnitfa: MACMILLAN AND CO., Ltd. [All Rights reserved.] THE ALGEBEA OF INVARIANTS BY J. H. GRACE, M.A. FELLOW OF PETEllIIOUSK AND A. YOUNG, M.A. LECTUKER IN MATHEMATICS AT SELWYN COLLEGE, LATE SCHOLAK OF CLAKE COLLEGE CAMBRIDGE: AT THE UNIVERSITY PRESS. 1908 Cambrtligc : PRINTED BY J. AND C. F. CLAY, /^'aK the ONlVE^Slit; PB'BJS. ". / Engineerinfj & Mathematical Terences Library CO CD CO 'StOl P PREFACE. ri^HE object of this book is to provide an English introduction to the symbolical method in the theory of Invariants. ^ It was started as an attempt to meet the need expressed by ^;^ Elliott in the preface to The Algebiu of Qualities — 'a whole book which shall present to the English reader in his own language Nj a worthy exposition of the method of the great German masters ^ remains a desideratum.' Since then the need has been partly ^ met by the article 'Algebra' by MacMahon in the Supplement v^ to the Encyclopcedia Britannica. The subject has been treated from the commencement in order that readers unacquainted with Elliott's treatise or any presentation of the elements may 4s be able to understand the argument. Such readers should bear fM^ in mind that this treatise is only concerned with one part of a very extensive subject. The modern theory of Partitions will „ be found in the first part of the article by MacMahon mentioned « above. V The first six chapters — a great portion of which, we hope, will %- be found easy reading — may be said to lead step by step to N^j Gordan's wonderful proof of the finiteness of the system for a single binary form. The sixth chapter is, in fact, devoted to an exposition of Gordan's thii-d proof, but here, as throughout the book, we have allowed ourselves a free hand in dealing with the memoirs and treatises quoted. For example, we have made much use of Jordan's great memoirs on Invariants in proving Gordan's theorem: in a later chapter on Types of Co variants the development of Jordan's method has led us to some results which we believe 2609G3 VI PREFACE to be important as well as novel, notably to an exact formula for the maximum order of an irreducible covariant of a system of binary forms. The remainder of the book is mainly of geometrical interest : much space is devoted to Apolarity and Rational Curves, and the treatment of ternary forms is from the geometrical rather than the analytical point of view. The only complete system of ternary forms given is that for two Quadratics: it may be felt that more should have been said on this subject, but we think that with the methods known up to the present the treatment of ternary forms is too tedious for a text-book. The number of references to Mathematical Journals etc. will perhaps be found unusually small : for this there is no need to apologise since the admirable Bericht ilber den gegenwdrtigen Stand der Invariantentheone* of Meyer gives references up to the last few years and in a more complete fashion than is desirable in a book which makes no pretensions to being exhaustive. We wish to thank Dr H. F. Baker for help given to us in our early reading and Professor Forsyth for encouragement while writing. For reading of proof-sheets we are indebted to Mr J. E. Wright, B.A., of Trinity College, Mr P. W. Wood, B.A., of Emmanuel College, and in a still greater degree to the late Mr A. P. Thompson, B.A., of Pembroke College, whose Enthusiasm for Mathematics and research was most helpful and whose early death is deplored alike by his teachers and his fellow-workei*s. Our thanks are also due to the officials of the University Press for great help received during the course of printing. J. H. GRACE. A. YOUNG. * Jahresbcricht der Deutschen Mathematiker Vereinitfunrj, Vol, i., 1892. French translation byFehr; Gauthier-Villars, Paris, 1897. Italian translation by Vivanti; Pellerano, Naples, 1899. Article, Invariantentheorie in the Encyclopadie der iiMthcmatischen Winsenschaftcn. August 18, 1903. CONTENTS. CHAP. I. Introduction PAQE 1 II. The Fundamental Theorem 21 III. Transvectants .... 36 IV. Transvectants {continued) 53 V. Elementary Complete Systems 85 VI. Gordan's Theorem .... 101 VII. The Qcintic 128 VIII. Simultaneous Systems . 158 IX. Hilbert's Theorem 169 X. Geometry 183 XI. Apolarity and Rational Curves 213 XIT. Ternary Forms .... 246 XIII. Ternary Forms (coniinued) . 274 XIV. Apolarity (contimied) 299 XV. Types of Covariants 319 XVI. General Theorems on Quantics . 339 Appendix I. The Symbolical Notation 365 )j II. Wronski's Theorem 370 >» III. Jordan's Lemma 375 Ii IV. Types of Covariants sidex . 378 . 381 4 CHAPTER I INTRODUCTION. SYMBOLICAL NOTATION. 1. If in the expression we write Xi = f^Xi + t/i X2 , W2= ^2X1 + 7)2X2, we obtain a new expression, viz. AoXi' + 2A,X,X2 + A.,X2^ where ^0 = flo ^1^ + ^(h ^1 f 2 + ^2 ^2', -4i = ao|i»7i + «! (|i7;2 + |2'7i) + a2^2'72, A2 = fto'/i^ + '^(hViV2 + (hvi- It is easy to verify the identity A^A^-A^'^ (aoUo - ai") (li?/, - ^^Vif, which shews that the function AqAs — A^^ of the coefficients of the transformed expression differs from the same function ao<*2 — ^i" of the coefficients of the original expression by a factor involving only the coefficients contained in the transformation. 2. In the present work we shall give an account of the theory and structure of functions of the coefficients possessing properties analogous to that described above ; but before pro- ceeding to generalities we shall give some further examples. If we transform the two expressions ^01 •" ^Otiwju/2 "T" CLnOCi^ f (^0 ^1 I ^(l\ X-yX^ T 0-2 X2 ) G. & Y. 1 2 . * THE ALGEBRA OF INVARIANTS [CH. I in the same way as before, and they become A;Xi^-\-2A,'X,X, + A,'X,\ then it is easy to verify the identity AoA^' - 2A, A,' + Ao'A^ = {a^a^ -2a^a^'-\- a^a^) (^^^2 - ^^ViY- Thus we have here a function of the coefficients of two ex- pressions such that the new value differs from the original value by a factor depending only on the transformation employed 3. As a third example, if the cubic expression become A,X^' + ;Mx X;'X^ + ^A^X, Xi + A^Xi, when we put a\ = l^jXi + t^iXj, then we have (^0^2 - A^) X^ + (^0^3 - ^1^2) ^1X3 + (^1^3 - A^) Z/ This identity indicates a property quite similar to that illus- trated in the two previous examples, but the function, which is unaltered except for the factor (^i?72 — ^2'7i)^ now involves the variables as well as the coefficients of the expression from which it is formed. The result we have written down may be verified directly, but more easily as follows : Denoting the original expression by f and the transformed expression by F we have to prove that aAvaZa* \dx,dxj \dx,^dx^^ Kdx^dxJ]^^'^' ^'^'^^ dF__dF_dx^ dF_dx^ dXi dxi dXi dx^ dXi '^'dx^'^^^dx^' 2-4] INTRODUCTION. SYMBOLICAL NOTATION 3 and in like manner d^F d^F d^F d^F d^F d^F d^F d^F d^F ,d'F _ d^F ?'F But these equations are exactly the same as those which express A^, A^, A2 in terms of Uo, a^, a^ (§ 1), hence The expression ^^ ^7^ — 5~ra ) ^^ called the Hessian of/. 4. Let us now explain the phraseology in common use when dealing with questions such as arise in our subject. Quantics. A rational integral homogeneous algebraic function of any number of variables Xj, x^, ... x^y is called a quantic. The degree in the variables is called the order of the quantic, and according as the number of variables is two, three, four v.'e call the quantic binary, ternary, quaternary Thus a binary quantic of order w is a rational integral homogeneous algebraic function of two variables which is of the nth degree in those variables. Such a quantic might be written but we shall find it invariably more convenient to write it i.e. with binomial coefficients prefixed to the various a's. The former of these expressions is now commonly written Vflo> ^1) *2> •••» (^nyj^\i '^2) > and the latter {a^, a^, a^, ..., an\xi,x^^, a very convenient notation introduced by Cayley. 1—2 4 THE ALGEBRA OF INVARIANTS [CH. I The mere consideration of the transformation of the binary form will be sufficient to convince the reader of the advantage of the introduction of binomial coefficients. Passing now to the case of any number of variables, we call the quantic a p-ary ^-ic when it is homogeneous and of degree q in /) variables. Thus the most general ternary quadratic is written and in general the ternary n-ic is written where the summation is extended to all values of j9, q, r satisfying the equality p + q + r = n. It will be noticed that here we have prefixed multinomial coefficients to the as. 5. Linear Transformations. The equations X.i = ^2^1 + '72>X^2 are said to constitute a linear transformation from the variables x^x^ to the variables X^X^ — it is of course implied that the coefficients on the right do not involve either set of variables. The determinant is called the determinant of the transformation. If D vanishes it is evident that x^ and x^ are virtually identical, for their ratio is constant, and hence, as the variables are always supposed to be independent, we shall throughout only deal with transformations which have a non-vanishing determinant. On solving for X^X^ we find Xi = {7)^Xi-'r],X2)/D * Xi = {-^2Xi + ^iXi)II> 4-6] INTRODUCTION. SYMBOLICAL NOTATION 6 SO that the passage back from the new variables to the old is eflfected by a linear transformation. This is called the inverse of the original transformation ; it is evident at once that its determinant is equal to ^ . 6. Let us now regard a linear transformation as an operator, which acting on a;^, x^ changes them to X^, X^, and let us consider the effect of two such operators acting successively. If the coefficients of the first are and those of the second y I ' . f- / / ?1 , ''71 , ?2 > '72 , then we have a^i = ^i^i + »7i^2) ^2=^2^1 + »?2-X'2J ' -X'l=^l'^l' + ^l'Z2'| X2=^2X^ ■\-'q^X^\ ' and the effect of the two operators acting successively is to change from the variables cci, x^ to X/, X^. Now on elimination of X^ , X^ we find ^i = (111/ + % la') X^ + (Iit;/ + -^iT?/) X^, ^2 = (I2I1' + ^7212') ^1 + (hvi + V2V2) X^'. And accordingly we can pass directly from the original to the final variables by means of a single linear transformation which we shall call 2. If we call the two preceding operators S and 8' we may write X = SS' and 2 is called the product or the resultant of S and S'. It must be carefully noticed that the order of the factors S and S' is essential in considering their product. In our example we supposed that S acted first and then S'. If S' had acted first and then S we should have X = S'S and it is manifest that S and S' are not in general the same. 6 THE ALGEBRA OF INVARIANTS [CH. I Since the resultant of two or any number of linear trans- formations is another such transformation, the whole set of linear transformations obtained by varying the coefficients is said to form a group — a continuous group because the coefficients ^ and 17 may be sxipposed to vary continuously. The determinant of 2 is equal to the product of the determi- nants of S and 8', as follows from the multiplication theorem for determinants. The product of a transformation and its inverse is a trans- formation which does not affect the variables, i.e. it is which is called the identical operator. The determinant of this is unity, and, as we have pointed out, the product of the deter- minants of a transformation and its inverse is also unity. 7. The idea of a linear transformation admits of immediate extension to any number of variables x^, Xi,...Xp and now the transformation consists of n equations Xr='^nX-^-\-^nX^+ ... + ^rpXj,, r=\, 1,...p. The determinant D formed with the |'s for elements is called the determinant of the transformation, and inasmuch as when D vanishes there is a linear homogeneous relation between the x's, we exclude as before all transformations having a vanishing determinant. If Z) 4= we can solve for the X's in terms of the x& and, as can be easily seen, each X is a linear function of x^, x^, ... Xp, so that we have Xr = Vn^i + Vr2^2 + • • • + Vrp^p> a linear transformation which is the inverse of the preceding one. As in the case of two variables, the resultant of two linear transformations S and T is a third linear transformation and on examining the coefficients in S it will be seen at once by the multiplication theorem that the determinant of % is the product of the determinants of /S and T. 6-10] INTRODUCTION. SYMBOLICAL NOTATION 7 8. In the earlier portion of this work we shall deal almost entirely with binary forms, and although we shall be constantly considering linear transformations and their etfects, yet the fact that they form a group will not be explicitly used. Our only object, in introducing these elementary properties of groups, is to point out that the connection between invariants and groups is intimate and universal — in other words, that every group has its accompanying invariants and, conversely, every set of invariants belongs to a group. 9. Invariants of Binary Forms. If a binary form / be changed by a linear transformation into a new form F, and a function / of the coefficients of F be equal to the same function of the coefficients of / multiplied by a factor depending solely on the transformation, then I is called an invariant of the binary form/ Thus for example in § 1 the identity (^0^2 - ^i') = (ao«2 - ai) (^iV2 - ^2Vi)" shews that a^a.^ — a-^ is an invariant of the binary quadratic An exactly similar definition applies to a joint invariant of several binary forms, e.g. a^a^ — ^a^tti + a^a^ is an invariant of the two binary forms and adx^ + '2a-lx■^X2 + a^x^. 10. For the present we shall confine our attention to invariants which are rational integral functions of the coefficients. It is easy to see that there is no further loss of generality if we suppose the invariants to be homogeneous in each set of coefficients that they contain. Thus for example if / be an invariant of a single binary form f which is not homogeneous in the coefficients a we can write / in the form /i + /a + . . . + /«, where each element in this sum is homogeneous. 8 THE ALGEBRA OF INVARIANTS [CH. I Now by definition we have I{A) = MxI(a), and therefore I,{A) + I,{A) + ... + Is(A) = M{I,(a)+I,(a) + ...+I,(a)}. But the A's are linear functions of the as and M is independent of both, and therefore the only part on the left-hand side which is of the same degree as I^ (a) on the right-hand side is Ii{A); .'. h{A) = Mh{a), that is to say I^ is an invariant. Hence a non-homogeneous invariant is the sum of several homogeneous invariants. This result can be at once extended to any number of binary forms. As an example ttoOs — a^ + a^a^ — 2a^a/ -I- a^ao is an invariant of the two binary quadratics (tto, Oi, as 5^1, ^2? and (ao', a/, a^'^Xi, x^f, but it is the sum of two expressions ttotta — tti", and aoOa' — 2aia/ -I- a<iao each of which is homogeneous in the two sets of coefficients. 11. Covariants of Binary Forms. If a binary form / is changed into a form i'^ by a linear transformation, and a function C of the coefficients of F and the new variables Xi, X^ be equal to the same function of the coefficients of /and the old variables ajj, a;, multiplied by a factor depending only on the transformation, then C is called a covariant of the binary form. Thus from what we have seen ay ay ( df Y dxi^ dx^ \dxi dx^ j is a covariant of the binary cubic/ and in fact of any binary form. An exactly similar definition applies to a joint covariant of several binary forms — as an example the reader will have no difficulty in shewing that the Jacobian §/;a5^_a/a0 dxi dx^ dx^ dxi 10-13] INTRODUCTION. SYMBOLICAL NOTATION 9 of any two forms / and ^ is a covariant of those forms, the multiplier being (^1% — |^2^i) the determinant of the transformation. We shall confine our attention to covariants which are rational integral functions both of the coefficients and the variables, and, as in the case of invariants, there is no difficulty in seeing that there is no further loss of generality in supposing such covariants to be homogeneous in the variables and in each set of coefficients involved. In fact if a covariant be not homogeneous it is the sum of several parts each of which is a covariant and homogeneous. 12. Degree and Order of a Covariant. The degree of a covariant of a single form is its degree in the coefficients of that form — the order is the degree in the variables. The covariant -~^ ^^ — f ^ ^ ) of a binary form of order n is of degree two and order 2n — 4. A covariant of several binary forms has a definite partial degree in each set of coefficients involved and the order is as before the degree in the variables. The Jacobian of / and <^ is of degree one in the coefficients of each of the two forms, and its order is the sum of the orders of f and (f) diminished by two. 13. Symbolicar Notation. In our investigations we shall find it of the utmost value to write the binary quantic ttoXi^ + naiXi^~^ 0)2+ ... +{ j a^a^i""'' a;/ + . . . + a^iPa" in the symbolical form so that tti" = tto, tti"""^ a.2 = ttj , . . . a^"-'' a/ = ar,... ^^ = an. This representation is startling at first sight, but consider how the use of it would introduce errors into calculation. They would arise because relations of the type ana» = a 2?l— 2 ~ 2 /, 2 between the coefficients prevent our binary form from being a general one. Now in representing a function of the coefficients 10 THE ALGEBRA OF INVARIANTS [CH. I symbolically we allow no symbol such as a to occur more than n times in any one term, so that the possibility of relations giving rise to aotta = O]^ is entirely precluded. In fact to orbtain this relation there must be 2n a's multiplied together in the representation of the function aorta or al^ whereas, when we allow no more than n a's to occur in any one term, the (n+1) expressions aA ai"-ia2,...ai"-'-a2'-, ...aa" are independent quantities, i.e. with these restrictions on the use of our symbols the {n + 1) coefficients of the original quantic are not necessarily connected by any relation, and therefore the most general quantic can be represented in the form indicated. Accordingly in addition to the symbol a we introduce a riumber of equivalent symbols /3, 7, ... so that /= (aia^i + a^x^)'^ = (^^x^ + ^^x^Y = (y^x^ + y^x^Y =... or as it will invariably be written J — Clx—Px—'Yx ••• The symbolical equivalent of aorta is not because here there are more than n a's multiplied together. To represent rtort^ we must use two different symbols a, yS and then aoaa = ai"/8i"-^A'> which is of course equivalent to /9l''a,"-2a2^ whereas in the same symbols a,'' is represented by oii^~^okfii^~^^2' In general to represent an expression of degree m in the coefficients, we have to use m different symbols of the type a, A 7. •••• We have said that not more than n a's must be multiplied together in a given term — on the other hand if the expression has an actual as well as a symbolical significance not less than n of these symbols must occur together because only the expressions ai", tti'^-^aa, ... Oa'' have an actual meaning. 13-14] INTRODUCTION. SYMBOLICAL NOTATION 11 14. A function of the coefficients can generally be represented symbolically in different ways as we have seen in the case of aoa, for example, which is equivalent to both an^n-2^2 and ^^^a,-^-^. There is one method of determining the symbolical repre- sentation which is very convenient because it often leads to the expression most suitable for our purpose. Suppose, in fact, that P is a homogeneous function of the mih degree in Uq, a^, ... an, then is only of degree m— 1 in a^, ai, ... an. If in Pi we replace each b by the corresponding a we obtain mP, as follows from Euler's Theorem relating to homogeneous functions. In like manner if in A i_Vfo~4-&— +b -)P ^9ai "" '^dajx'^dao ^ dai '" ^daj we replace each c and each b by the corresponding a we get m (m — 1) P and P^ is of degree m — 2 in the a's. Proceeding in this way we can find an expression Pm-i which is linear in each of m sets of symbols a, b, c, ... k, and which becomes equal to P x ml when each b, c, ... k is replaced by the corresponding a. Now having formed the expression Pm-i we replace each a by the symbol a, each b by the symbol yS, each c by the symbol 7 and so on. Since the expression is linear in each set of letters, each symbol will occur exactly n times in every term, and then, regarding the symbols as referring to the same quantic, we have the required symbolical expression. Thus for example aoa, - a,^ = i (b, ^ + b, ^ + 6, Aj («„«^ _ a,%^^ = i (&o«2 + &2«o - 2ai6i)j=a = i (AW + ^,W - 2aia,,8i A) a,-'^,^-' 12 THE ALGEBRA OF INVARIANTS [CH. I and the convenience of this expression in terms of a, y8 will be abundantly evident in the sequel. Ex. (i). For the binary quartic shew that and flo «i «2 | = H«i/32-«2/3i)^Oiy2-^2yi)^(yia2-y2«i)*- Ui a^ ct^ I ttg CTg ^4 I Ex. (ii). By the same method shew that for any binary form Ex. (iii). Shew that for a binary form of odd order (a^^2 - cii&i)"' is zero and write down its value for a form of even order in terms of the coefficients. 15. Polar Forms. The expression (n-r)\f d d y^^-^y-' -Y/ where y*= aa;" = )Sa;'' = etc. is a binary form of order n, is called the rth polar of/ with respect toy. The operator ( t/i ^ — \-y^K-\, which is frequently written l^'-a^. + ^'a^J-^ is said to be derived from / by polarizing r times with respect to y. The numerical factor ^^ -^ is only introduced for con- venience. These polar forms admit of very simple representation in our symbols, for and so on. Hence the rth polar of / with respect to y is (n — r)! w! w (71 - 1) . . . (n - r + 1) (x^-''tLy\ that is OiJ^-^'aJ. 14-16] INTRODUCTION. SYMBOLICAL NOTATION 13 The differential coefficients of f with respect to the variables are particular cases of polar forms. For if 2/1 = 1, 2/2 = 0, the rth polar is ay aa;/ and if y-^ = 0, 3/2 = 1, the rth polar is 9^2""' In general we have aP+«/" n ! J^ — /y n-p—qn Pn 9 ^x^-dxS {n-p-q)\ "" ' '• The form ^ ^ [y^] (•^5~)/^^ called a mixed polar with respect to y and 2 ; its symbolical expression is a^n-p-q^a^q^ 16. Effect of a Linear Transformation. If we write ^^1 = ^1^1 4- 771X2 ^2 = ^2X1 + Vi^z, then Ux becomes or ajZi + a^Xg, and hence the binary form a^^ becomes (afZi + a,Z2)" or ai" Xi" + n aj'*-^ a, Xi"-^ X^+...+ a^X^. Accordingly in the transformed expression the coefficient of Xj** is found by replacing x by ^ in the original form, and the coefficients of Xi'*~^X2, X-^~'^X} ... are found by polarizing the coefficient of X-^ with respect to i) once, twice Of course suitable numerical multipliers must be introduced. The reader will easily illustrate this result by reference to the transformation of a binary quadratic in § 1. 14 THE ALGEBRA OF INVARIANTS [CH. I = «!, "2 X ^1, ^. ^1, A Vi, V2 17. The form a^^ = ^x^ = 7a;" • . . becomes on transformation (a^X, + a,X,r = (^f Z, + /3,X,r = (y^X, + y.X.y =.... Now we have All + Af2, /3l»7l + A'?2 = («! A - otsA) (^i'72 - fa'/i), a result of fundamental importance. We shall denote the expression (ai/Sa — tto/Si), which we call a symbolical determinantal factor, by (a/3), so that (ay3) = — (/9a), and the above relation may be written af/g,-a,;8i = (a/3)(^^). To illustrate these remarks let us prove that do 0^4 — 4aia3+ Soa^ is an invariant of the binary quartic We have (00^4 - ^cb\(h + 302=*) = :| f 26 5- j (ttotti - 4aia3 4- ^a^)h=a = i (a/8)*. Thus, if the coefficients of the new form be denoted by capital letters as usual, we have (^0^4 - 4^i^3 + 3^2^) = i (aj A - a, A)* as follows from the symbolical expression given above. But since a^ ^, - a,ySf = (a/9) (^17), A^A^ - 4^i^3 + 3^2^ = {^tjY (aoa^ - ^a^a^ + ^(h% which shews that a^ai— 4iaiaz + Za^ is an invariant and that the multiplying factor is the fourth power of the determinant of transformation. 18. Symbolical expressions representing Invariants. If the symbolical equivalent of an expression /, homogeneous and of degree % in the coefficients of the binary form be an aggregate of terms each of which is a product of factors of the type (ay6), then / is an invariant of the quantic. 17-19] INTRODUCTION. SYMBOLICAL NOTATION 15 For let I = ^T, where T is the product of w factors of the type (ajS), then the total degree of T in the symbols is 2w and it is also n x i, for there must be i sets of symbols, each set occurring to degree n\ therefore ni=2w, so that w is the same for every term in the aggregate representing /. If /' be the same function of the coefficients of the transformed expression, then /' = ^T' where T' is found from T by replacing a^ by a^ , a.2 by a, and so on. But since (a^/S, - a,/3f) = (a/S) {^t}) it follows at once that T = {^y T, and therefore I' = {^7))^ I since w is the .same for every terra. Hence / is an invariant. Exactly the same result is true for any number of binary forms if we suppose that / is homogeneous in each set of coefficients, for it is easily seen that the number of determinantal factors must be the same in every term, it being in fact when Til, n^, ... are the orders of the forms and i^, i^, ... the respective degrees of / in the coefficients of the forms. The rest of the proof then depends only on the fact that, whatever a and /3 are, we have Thus / is an invariant and the multiplying factor is now 19. This simple theorem enables us to construct as many invariants as we please — we have only to write down a product of factors {a^) and take care that the symbol a occurs in n of these factors where n is the order of the form to which the symbol a belongs. If this condition be not satisfied the invariant property still holds but the expression has only a symbolical meaning. On the other hand, if every symbol occur to the right degree but the expression be not reducible to the form above, it is an actual function of the coefficients which is not an invariant. 16 THE ALGEBRA OF INVARIANTS [CH. I As an example we have an invariant of the second degree (fliS)" for a binary form of order n. This vanishes identically when n is odd, as can be seen by expressing it in terms of the' coefficients ; or thus, since a, /3 are equivalent symbols and hence (a^)" = (-l)"(ayS)", giving the result at once. Again, for the binary cubic we have the invariant and for the binary quartic the invariants (a/3)^ (a^n^yYiyay, {a^y(ay)(^8){y8y. In every case it will be observed that the multiplying factor is a power of {^v)- As an example of invariants of several binary forms we may mention (ayS)", an invariant of the two different binary forms fla:" and /9a;". For quadratics this is the well-known invariant of §2. Again (a/3) is an invariant of the two linear forms a^ and ^^ and in this case a, /8 are actual coefficients as well as symbols. Then (ayS) (ay) is an invariant of the quadratic Ox^ and the linear forms /3a,, y^. 20. Covariants. A similar method exists for constructing covariants. Commencing with an example let us prove that the Hessian ^_ayay /ayy dxi^ dxi \dxidxj is a covariant of the binary form /= a^^ = /3a;" = . . . Since H is of the second degree in the coefficients ^ V daJ Ida;,' dx^^ \dx,dxj j b=a \dxi^ dx^ "dx^ dx^ dx^dx^ dx^dxj 6=a where /' = {\h^ . . . bn$XiX^y\ 19-20] INTRODUCTION. SYMBOLICAL NOTATION 17 Replacing the as by as and the 6's by /3's as usual, we have H=^n'' (n - 1)2 {ai^a^'^-^ye^^yga.'^-a + a.^a,-^ A^^^-2 - 2a,ct,a^"-^ A/82/3«,"-='j as can be immediately verified by expressing this in terms of the coefficients. The transformed quantic is and the corresponding expression derived from this is (a^/3, - a,/3|y^ (ajZ, + a^X^y-'i^^X, + ^,X,f-' which shews that the expression is a covariant and that the multiplying factor is (^■j/)^ In general, if an expression C, of degree i in the coefficients of/ and of order m in the variables, can be symbolically repre- sented as an aggregate of terms, each of which is the product of a number of factors of the type (a/S) and a number of the type a-t, then G is a covariant oif. In fact let C = SF, where F is such a product. The number of factors with suffix x in F must be m, the order of G, and if w be the number of the type (a/8) we have 2w + m = ni, for each of these represents the degree of G in the symbols. Hence lo is the same for every term. If C" be the corresponding expression derived from the transformed quantic, then G' = 2F', where F' is derived from F by replacing «! by a^, a^ by a, and so on, and «« by {a^X^ + a^Xg). Thus since (a^^, - a,/3f) = (a/3) (^t;) and a^ = a^Xj + a^Xj we have F = {^r}Y F, .-. C" = (|^)«'C, that is to say C is a covariant. G, & Y. 2 18 THE ALGEBRA OF INVARIANTS [CH. I Exactly the same method applies to a covariant of any number of binary forms, but now the symbols a, /S, ... may refer to different forms and, of course, a symbol such as a. must occur in the symbolical expression to the requisite degree. We can thus easily construct any number of covariants of one or more forms, e.g. for a binary form of order n is a covariant for any integral value of r, but it vanishes when r is odd. Again, if <x^, ^^ are two different quantics, is a covariant ; if r = 1 it is the Jacobian. As further examples we have the covariants (a/3)» {&if ina)- ct,;S,7„ (ay8) {^i) (7a) (aS) (^g) (7S) a^'^i^A^ of the binary quintic a.'' = y8.' = 7^» = a/=... As an exercise the reader may prove that the last one vanishes identically. 21. We have seen how useful the symbolical methods are in constructing invariants and covariants. In the next chapter we shall prove that they constitute an ideal calculus when we shew that every invariant and covariant can be represented as a sum of symbolical products of factors of the types (a/S) and a^. Meanwhile anticipating this result we shall indicate the methods of trans- forming symbolical expressions. These depend on two principles : (i) Interchange of equivalent symbols, (ii) Identities in symbolical expressions. According to (i) if a symbolical expression have an actual meaning and contain two equivalent symbols then its value is not altered by interchanging those symbols. We have already used this method in proving that the invariant (ayS)" of the quantic a^^ = /3^" 20-22] INTRODUCTION. SYMBOLICAL NOTATION 19 vanishes when n is odd. As another easy example we have (ay8)(/37)(7a) = for the quadratic a^^ = jSx^ = yj', or for the two different quadratics a^^ = ^^^ and yx^. More generally the co variant (ayS) (M (7^) ax''^^x''-V-' is alw^ays zero unless the three forms a^^, /3a;" and jx^ are all different. 22. Fundamental Identities. We have identically (Moix + (ya)^x + {ci^)yx==0 (I). as can easily be verified. From this identity many others may be deduced. For example, replacing cc^ by B2 and x^ by — Si we have (/37)(aS) + (7a)(;S8) + (a/3)(7S) = (II), a result useful in transforming invariants. Again from (i) (M ^x = (^a) Ix - (7«) ^a; and hence by squaring 2 (otyS) {ay) ^xlx = (a/3)-^ Ix' + {ayf ^x' - {^yf a,^ • • .(HI). As identities less generally used we may mention {^yf ax' + (7a)' ^x' + {oi^r Ix' = 3 (M (7a) (^'/S) aa=/3a=7* ()S7)'«.^ + (7«)^/Sa.^ + («;Sy7*' = 2 {(a;8)^ (a7)-^ /3,.^7,,^ + {^yf {^af yx^ai + (7«)' (7/^)^ a*^/3«=1. Ex. (i). For the quadratic (a^) (ay) /3,y, = ^ {(a^)^ y^ + (ay)^ ^,2 - {^yf a,^} since the symbols a, /3, y are equivalent. 2—2 20 THE ALGEBRA OF INVARIANTS [CH. I Ex. (ii). If fi = a^=^^ and fi — ^x=^x be two different quadratics, to express the square of the Jacobian J={aa)axa'x in terms of/ and /'. We have J^={aa') a^a'^ {m') ^x^'x or since (|3/3') a'^={^a') ^'^ - O'a') ^^ J^={aa') i^a') a,^,^J^- {aa') O'a') a,^'^ . /3,2 = i3',2 i {{aa'f ^,2 + Oa')2 a,^ - {a^f a',^} -^,m(aa'r^,^ + {P'ayaJ-{a^ya'J^} by (III), or if (a/3)2=/,i, (aa')2 = (a0')^=...=/,2, {a'^f = l^^, we have 2 J2 =/2 {/j^^/-^ + I^J^ _ /^^/J ~/i{m2/2 + A2/1 ~ ■'12/2} Ex. (iii). Prove that for the binary quartic (a^)2(ay)2^,2y^2=.|_^.(„^)4 (o^) (ay) a,2/3,3y^33=^y. („^)2 „^2^^2. CHAPTER 11. THE FUNDAMENTAL THEOREM. 23. It will be remarked that in every example of in- variants and covariants, discussed in the preceding chapter, the symbolical expression for such a function involved only factors of the types (a/3) and a^;, and further that the multiplier alluded to in the definition was always a power of the determinant of the transformation. We are now going to establish the general truth of these properties. As a matter of history, we may observe that the original definition of an invariant stated that the multiplier was of the form mentioned ; but following the logical, rather than the historical order, we shall first prove that the multiplier must be a power of the determinant and then proceed to prove the proposition relating to the symbolical forms for invariants and covariants. 24. Suppose that / is an invariant or covariant of a single binary form f — after what has been said, § 10, we may assume that I is homogeneous in the coefficients of/. Let the linear transformation change / into /' and let /' be formed from /' in the same way that / is formed from f\ then, by definition, /'=i^(^i,77i, f 2,772) x/ and we have to shew that F is simply a power of (f 1772 — ^2''7i)- 22 THE ALGEBRA OF INVARIANTS [CH. II Now let a second transformation Xi ^= q\Xi + 7Ji X^ X^ ^ 52 '^\ "■" '72 ''^2 change /' into /", and let /" be formed in the same way from f", so that Hence we have /" = ^(ri. '71. ^2, 1.) X F{^,\ 7;/, ^/, ,;;) X /. But we can pass from the variables x^, x.^ to the variables x", x^' by the single transformation Xi = (fi^i' + 171I2') a;," + (^i77i' + T/iW) a^a" a^2 = (^2^1' + '72I2') ^1" + (^2';i' + '72'?2') ^2" ; therefore Consequently F must satisfy the functional equation = F(^„ 71,, ^2, %) X F{^^, V. r;, ^72'). The solution of this equation is not difficult. In the first place we remark that since ^1 = 1, 7/1 = 0, ^2 = 0, % = 1 gives the identical transformation, F{\,0,i),l) = \. Again putting ^1 = k, 7/1 = 0, I2 = 0, '72 = ": each new coefficient is equal to the corresponding original coefficient multiplied by the same power of k, in this case the multiplier is clearly a power of K, i.e. F{k, 0, 0, /«;) = «:'■. Since F{^u Vi,h, V2) X F(k, 0, 0, k) = F(k^„ KVi, K^2, 'CV2), we have F(k^i, /CT/i, K^2, icVt) = K-^'Fi^i, 77i, ^2. '72), therefore F is homogeneous and of degree r in the four variables 1^1. ^71. 1^21 Vi- Finally let us choose ^Z, t;/, I2'. %' so that ^i|i' + 771^2' = 1 > |i»7i' + »7i'72' = 0. ^2^^/ + ^72^2' = 0, |2'7i' + '72^2' = 1. ^B 94-_9fi1 24-26] THE FUNDAMENTAL THEOREM 23 which relations give {D = ^j772 - ^27/3), then we have Fi^,, Vl, ^2, V2) X i^(^/, %', f/, 772') = i^(l, 0. 0, 1) Consequently since ^ is homogeneous and of degree r F (^1 , %, I2, '^a) X i' (772, -vi, - ?2, f ) = ^. But inasmuch as D is obviously irreducible — i.e. it cannot be resolved into factors — and F is clearly an integral function, this equation shews at once that both -^(^1. Viy ^2, V2) and F(r}2, - r)^, - ^2, ^1) are integral powers of D. Hence the theorem is established. 25. Assuming the truth of the proposition just proved, the proof of the fundamental theorem that invariants and covariants can be completely represented by factors of the types (a/8) and a^ is very simple in principle. The actual work requires two lemmas of great importance in the present subject, and we shall give them separately. Tbey are both concerned with properties of the differential operator 26. Lemma I. If n be a positive integer In fact ^— (x^y^ - x^y^Y = ny^ {x^y^ - x<^iT~^, + ?? (n - 1 ) (^13/2 - iio.jy^y-^'X{y^ 24 THE ALGEBRA OF INVARIANTS [CH. II Similarly 92 ^— - {x^y^ - x^,y =-n {x^y^ - x.y^f-^ ^n{n-V) (x^y^ - x^y.f-^x^y, . Consequently n (^i2/2 - x^y^Y ={n(n-l) + 2n} {x,y., - x^y.y-' = n{n+l) {x^y., - x^yif-\ which establishes the lemma. If we operate again with H we find fi2 (^x^y^ - x^y^Y = n{n + \){n-\)n{xiy^- x^y^f-'' and in general = (n + l)n'{n-iy ...{n-r + 2)^ {n-r + 1) {oc^y^-x^y-,Y-^ or n*- {xyY = (n + 1) /i^ {n-iy ... {n-r + 2)^ {n-r + 1) {xyy^. Finally n« (a^i 2/2 - ^22/1)" = (n + 1) (w n^' a constant which is not zero — for our immediate purpose this is the important result, and it can be at once verified by expanding \dx,dy~ dx,dyj ' ^^^2/2-^2^1) by the Binomial Theorem. 27. Lemma II. If the operator fl be applied r times to the product of m factors of the type a^. by n factors of the type fiy, then each term in the resulting expression contains r deter- minantal factors {ol^), (m — r) factors Ox and {n — r) factors ^y. To ensure perfect generality we consider where P = Oa,*" aa,<''' . . . a^;*'"' and Q = /3/)^/'.../3,<»', the a's and the ^'s being all different. H 26-28] Wm Now 26-28] THE FUNDAMENTAL THEOREM 25 where the summation extends so that r takes all the values 1, 2, ... m and s takes all the values 1, 2, ... n. Hence on subtraction P Q nP.Q = SK)/3'«0.,,,^„. which establishes the lemma for r = 1. But since the operator CI has no effect on a factor of the type (a"''yS<**) the theorem holds for r = 2 ; in fact and performing the operation on the right we have the result. Proceeding in this way we see that at each step a new factor of the type (a/S) appears in each term while one factor of each of the types ax and /3y disappears — this completely establishes our lemma. Ex. (i). Prove that Q'' a '" 6/ = , , , , ^ (ahY a^-"" 6 "-^ Ex. (ii). With the notation of the text prove that Q'' F. Q contains every term of type there written r ! times and that the number of different terms is , '~^, J — ". , — , . (Use induction.) (m — r)! («.-r) ! r! ^ ' 28. Fundamental Theorem. Suppose now that P(ao, ai, ... a„) is an invariant of the binary form (tto, ai, ... a„][^i, x.^"" = a^" = ^^« = ..., then after the linear transformation ^2 = ^2^1 + •J72^! 26 THE ALGEBRA OF INVARIANTS [CH. II we have seen that a^ becomes aj and a^ becomes a, ; so that if the new form be we have Ar = aj^^^a/. By definition F(Ao, A„ ... Ar,) = (I1172 - hv^yFia,, a„ ... a„) ; accordingly if the A's are replaced by their symbolical expressions F becomes the sum of a number of terms, say XP . Q, wliere P contains only factors of the type a^, and Q only those of the type a,. As the degree in ^ and rj must be w we infer that there are just w factors in P and w in Q. If we operate on both sides with D,'^, P . Q becomes the sum of a number of terms each of which is the product of w factors of the type (ayS), and the result on the right-hand side is a numerical multiple of J^(ao«i ... ««)• Hence we have expressed F(ao, ai, ... an) in the symbolical form peculiar to invariants. The proof as given applies to invariants of one binary form ; it is the same, word for word, for any number of binary forms, for the left-hand side is still of equal degree in f and rj, and on the right-hand side we have the determinant {^r)) occurring to a power equal to this degree. Hence operating as above the required symbolical expression is obtained. 29. The proof for covariants is of the same nature as that for invariants, although a little more care is required in the manipulation of the symbols ; after what has been said on in- variants we may confine our attention to covariants of a single form. Suppose that F{ao, a^, ... an, oci, x^ is a homogeneous covariant of order m of (tto, Oi, . . . ttn^iCi, X^y = Oa,^ = /8/ = etC. Then using the same notation as before F{Ao, Au ... An, X^, X^) = (^iV^-^-iViyPicio, ai. ••• ««, ^i, ^2). 28-30] THE FUNDAMENTAL THEOREM 27 If the A's are replaced by their symbolical expressions we get where P involves only factors of the type a^ and Q only those of the type a^. But on solution we have fl'72-t2'7l Cl»72-?2% Now for convenience we shall replace x^ by Wg s-nd a;^ by — Mi, so that X, = ,, "% , , Z, = - Wf (|l'72 - ^2'7l) ' (|l';2 - f2'7l) * Substituting these values and multiplying up by (^rj)^ we obtain the identity 2 (- 1)"*= P . Qu^'^iu^'"^ = (^t; )«-+"» i^. We may write the left-hand side XP' . Q' where P' only contains factors with suffix f and moreover exactly (w + m) factors, and Q' contains {w + m) factors with suffix 77. Accordingly after operating with 11^+*" each term will involve {lu + m) determinantal factors, and the right-hand side will be a numerical multiple of F. Now of the {w-\- m) factors, there are w of the type (ayS) and m of the type (aw), for u must occur to degree m in the final as well as in the original expression. But {au) is — OLx ', hence replacing the it's by the xb throughout we have the symbolical expression for F. 30. Since we have proved that all invariants and covariants of one or more binary forms can be completely represented by products of factors of the types (a/8) and ax, and further that every expression which can be so represented is an invariant or covariant — provided it possesses an actual significance, — it follows at once that all properties of invariants and covariants are implicitly contained in the symbolical representation and can be deduced therefrom. 28 THE ALGEBRA OF INVARIANTS [CH. II 31. Let us examine somewhat more closely the constitution of invariants of a single binary form /= (tto, tti, . . . an^xi, Xof = OL,'' = ^x" = 7*" etc. Suppose that an invariant / is an aggregate of products of factors (a/S) such that in every term there are w factors, then inasmuch as each symbol occurs n times in / we must have ni = 2w, where i is the number of different symbols. Now the weight of a,, is r by definition and its symbolical equivalent is ai^'^a/; hence the weight of any product of the as is the sum of the weights of the factors and is therefore equal to the total degree to which the letters ««, ySg, 72, ... occur in the symbolical equivalent. In the case of an invariant such as / each term in the symbolical expression when multiplied out is the product of w symbols with suffix 1 by w symbols with suffix 2 ; hence the weight is w. Further, the multiplying power of the determinant for I is also w. Consider next a covariant of degree i and order m. It is an aggregate of terms, each of which is the product of the same number (say p) of factors of the type (a/8), by the same number (say q) of factors of the type Ox. We deduce at once the relations q = m, 2p + q — ni, for each member of the latter equation represents the total degree of the covariant in the symbols a, /S, 7, Thus m = q = ni— 2p. 32. The leading coefficient of the covariant is called a seminvariant — it is found at once from the covariant by putting cci = l, X2 = and therefore represented symbolically it is an aggregate of products of p factors of the type (a/3) by q factors of the type ai. 31-33] THE FUNDAMENTAL THEOREM 29 The weight of this seminvariant is accordingly p, and hence we infer that if w be the weight of a seminvai'iaut and i its degree the order of the corresponding covariant is ni— 'ho. Thus for example in connection with the cubic we have the covariant {a^ya^^x- The seminvariant is i.e. its weight is 2, as we should have inferred from the number of factors (olIS) in the covariant. The order of the covariant is 2 and here we have i =2, n — 2, 7n= 2, w = 2, so that m = ni — 2w. In like manner if the leading coefficient of a joint covariant of two quantics of orders Jij and n^ be of degrees i^, 4 in the respective coefficients and of total weight w in these coefficients conjointly, then the order of the covariant is riiii + 7?2t2 — 2w. The reader will readily establish this theorem and extend it to the case of any number of quantics by using the symbolical notation. On putting 771 = we get a relation connecting the degrees and weight of an invariant. 33. Deduction of a covariant trova. its leading coeJBEicient. As we have seen, each term in the symbolical expression for the seminvariant must be the product of w factors of the type (a/9) by m factors of the type ai- Now suppose that in the seminvariant we replace ai by CLg, A by ^x, etc., and leave unaltered a., /S.,, etc., then (a/3) becomes (oiiTi + aoX^ A - (ySia?i + ^^x.^ tta = (a/3) on^ ; hence the seminvariant S is clearly changed into x^^ multiplied by the corresponding covariant — e.g. in the cubic, (a/3)"^ai/3i becomes xi^ X {oi^foix^x- We have thus a simple means of passing from the leading coefficient to the covariant. A similar result for 30 THE ALGEBRA OF INVARIANTS [CH. II invariants may be obtained by taking the particular case m = ; here the leading coefficient is of course the invariant itself. Let there be an identical rational algebraic relation among a number of seminvariants S^, S2, ... Sr and let G^, G^, ... Gy be the corresponding covariants. If the relation be and Wp be the weight oi Sp, then the sum must be the same for every term — hence if we put the left-hand side of the relation into symbols and then change a^ into ax, /9i into ^x as above, we have or 2(7j'*'C/»...a/' = 0, i.e. the covariants are connected by the same relation as the seminvariants. 34. Again when we replace Ui by Ox and leave a^ unaltered we replace the coefficient a^ by ax^-^OLr= _( n-r)l 9^ n! dx/ Hence except for a multiplier, which is a power of x, a covariant is the same function of / 1 _§t 1 dY {n-ry. ay i ay as the corresponding seminvariant is of ©0 , Ctj , £12 > • • • ^r > • • • ^n • Ex, (i). If in a seminvariant of weight w we replace ag by a^, /Sg by ^x, etc., and leave aj , ^^ . . . unaltered, then the result is the seminvariant multiplied by ^Tg". What is the corresponding transformation of the actual coefl&cients ? Ex. (ii). Find the result of replacing a^ by a* , og by a,, , ^i by /Sf , /Sj by ^T, , etc., in a seminvariant, and give the corresponding transformation of the coefl&cients. 33-35] THE FUNDAMENTAL THEOREM 31 Ex. (iii). Extend all the above results to the case of two or more binary forms. Ex. (iv). Prove that if in an invariant of a single binary form a^ be (n-r)\ d^f replaced by 7-^ ^—^ and so on, the result is the invariant multiplied by (n-r)\ d^f x^. State the result of replacing a^ by -^ j-^' ir-^ and extend the argument to any number of binary forms. 35. Alternative proof of the Fundamental Theorem — The Aronhold Operator. We shall now give another proof of the theorem of § 28 in which the original argument of Clebsch will be followed. Let be a covariant of a form /=(ao, tti, ... an][a?i, x^"" which is homogeneons in both the coefficients and the variables, and of degree i in the former. If F = {A„A,,...An\X,,X^- be the transformed quantic, we have </)(J.o, ^1, ... ^„) = /A</> (tto, ai, ... a„) where /x. depends only on the transformation. Now if (&o. ^1, ••• ^n][«i, ^2)" be a second form which transforms into then (tto + >^^o, tti + X6i, ... an + ^h^x^, x^"^ transforms into (^0 + Xi?o, 4i + \5i, . . . ^« + X5„][Z„ X^f . Therefore </)(A + ^5o, ^i + X-Bi, ... ^„ + \5,,) = yu,^ (tto + X6o, tti + X6i, . . . a» + A.6„), hence expanding by Taylor's Theorem and equating coefficients of X we have 32 THE ALGEBRA OF INVARIANTS [CH. II therefore the expression on the right is a joint covariant of the two forms ; in other words the property of invariance is not affected by an operator like . da) "" V ° 9^0 ^9^1 * * " " da J ' Hence, proceeding exactly as in § 14, we can construct a covariant of i different quantics which is linear in the coefficients of each, and which becomes a numerical multiple of (f> when we replace each of the i quantics hyf. The operator (&^) is called the Aronhold operator; its importance lies in the fact that it enables us to construct simul- taneous invariants or covariants of several binary forms of the same order when any invariants or covariants are known for a simple form. Thus, for example, since aoa-i—a^^ is an invariant of the quadratic (I/Q OC-y "7~ ^ (X\ w/j 00)^ "^ ttg U/2 the expression aJ)^-{-aJ)Q— 2ai6i is a simultaneous invariant of the two quadratics (tto, ai, a^x^, x^^Y and (6o, h, h^x^, x^\ The construction of other illustrations will present no difficulty. 36. It has been proved in § 14 that any covariant of degree i can be symbolically represented as a function </> of degree n in the coeflBcients of each of n different linear forms Oi, )3z> Jxy ••• '■> since ^ is a covariant of the original quantic it is unaltered by any linear transformation, except for a factor which depends only on the transformation, hence also it is a covariant of the i linear forms. By further use of the Aronhold operator we can now find a covariant of ni different linear forms a^W, a;,(2), ... ax(") )3:c(i), 0A ... /3x(») etc. linear in the coefl&cients of each form, and such that it becomes a numerical multiple of the original covariant when each of the symbols a(»), a(2), ... a(») is replaced by a, each of the symbols ^(1), /3(2), ... /3(») by /3, and so on. 35-37] THE FUNDAMENTAL THEOREM 33 We need therefore only consider linear covariants of linear forms in the sequel ; we shall prove that every covariant of a system of linear forms is a rational integral function of invariants of the type (a/3) and covariants of the type ax- Once this is established the general theorem follows immediately. 37. System of Linear Forms. First consider a single linear form and let (p be any invariant or covariant. If we use the linear transformation given by where b is any constant, then the new linear form is Xi and we have the equation where fi depends only on the transformation, i.e. only on a^, ag, b. Now let (f) = -\}rQai^ + ylria{'~^a2+ ...+'^r<i2' where the \|/'s do not depend on oj, ag but contain only x^, x^. Then *=%J/ + ^I'i^i'-M2+...+*r^2'' where ^^ is the same function of JSTj, Xg as x//-^ is of o^j, ^g > ^od further -4^ = 1, ^2 = since the transformed form is X^. Hence *o = M^ = /^(>^oai'' + ^iai''"^a2+---+^ra2'") (I)- Now ^0 depends only on X^, JTg, therefore it is of the form C,Xi'»+(7iXi™-iX2+... + (7^Z2-, the C"s being numerical, hence equating coefficients of x{^ in the equation (I) we find C^a{^ = iLX where X does not depend on b. Consequently /n does not depend on b and therefore on making 6 = in (I) we find CoXi™ = ;x<^, for X^ is now zero. Hence /x is constant and (/> is a numerical multiple of Xj"*, i.e. of Thus a single linear form has no invariants and the only covariants are powers of the form itself. We shall now assume that a covariant of any number, less than n, of linear forms which is linear in the coefficients of each form can be expressed in terms of invariants of the type (a/3) and covariants of the type ax. Let <^ be a covariant of the same nature of n linear forms ««» ^x) yxj 5*. ."J G. & Y. 3 34 THE ALGEBRA OF INVARIANTS [CH. II let 01 be the result of putting ^=a in (^, (^2 ^^e result of putting y = a in (f)^, and 80 on, so that 0„_i is the covariant of the single form Og obtained by making a = ^ = y=... in (j). Now since is linear in /3 we have and in like manner '^'^V'hi^"'^)'''' etc. Consider (a, g- + o^ ^l) <^ = 0i as a differential equation for (f>, it being given that 0i does not contain 3. As a particular solution we have where *S' is the degree of (^j in a, i.e. S= 2 in our case, therefore where O^ ) =^i^ +^29^, and further yj/^ is linear in ^. Accordingly ^^1 = ^1^1+ P2&2 » where F^, P.^ do not contain ^ and aiPj+a^P^^O, • ■ „ ^ ^1' 02 «i where xi does not contain /3 and is integral in a, y, 8, etc. Hence V'i = A^i+A^2 = («)3)xi and <^ = | (^ 9^)01 + («^);tl• But is a covariant, and ( ^ ^ j 0i is a covariant ; therefore {ai3) xi is a covariant ; again, (a^) is an invariant, therefore xi is ^ covariant. Moreover since xi(°^) i^ linear in a, ^, y, S, ..., xi is linear in y, S, ... ; therefore by hypothesis it can be expressed in tenns of (yS), y^, 8^, etc. Thus (f) = -(^^\(f)i together with an expression depending only on (a^), Qxy and factors of these types. 37] THE FUNDAMENTAL THEOREM 35 Then in like manner therefore * = 2^(^a-a) (^Da) ^^ together with terms of the required form. , Proceeding in this way we finally have *4(4)(ra^)('a^)-*. together with terms of the required form. But ^„ being a covariant of a^ is a numerical multiple of a." and therefore (^ 8^) (v ^) (4) * " ' '^" is a numerical multiple of a^^xyx •••• Thus we have expressed the covariant completely in terms of the two types of factors (a/3) and a^, that is, if the theorem is true for less than n forms it is true for n forms ; but it has been proved true for one form, hence it is true universally. Q. E. D. 3—2 CHAPTER III. POLARS AND TRANSVECTANTS. 38. Two sets of variables Xi, ^2 > • • • ^n ) yi,y^., ••• Vn, there being the same number of variables in each set, are said to be cogredient, if, when one set is transformed by any linear trans- formation, the other set is transformed by the same transformation ; thus if a?i, a^j, ... x^ become Xj, X^, ... X^ where •^1 ^^ ^1,1-Ai + 'i,2 A2 T ... "I n, n-An X2 = ^2, 1 -^ 1 > ^2, 2 -^ 2 "I" • • • I ^2, w ^ n .(I), then 2/1, 2/2, •^n — 'n, 1 A 1 + In, 2 A 2 4" ... "T l'n,n-^n yn will become Fj, Fg, ... Fn where 2/1 = ^1,1 -t 1 + ^1,2 ^2 + ••• H" n.n-'^n 2/2 == ^2,1 ^1 + %2 -''^2 H" • • • + ^2,n ^ n .(11). Two sets of variables Xi, X2, ... ^n 2/1.2/2, ... 2/n are said to be contragredient if, whenever the first set is transformed by the equations (I), then the second set of transformed variables Fi, Fg, ... F„ is given by the equations 1^1 = ^1.12/1 + ^2,1^2+ ••• +ln,iyn -» 2 = ^1,22/1 + ^2,22/2 + • . . + l"n,2yn l^n= ^i.n2/i + ^2,ny2 + • • • + In.n^n, .(III). 38-39] POLARS AND TRANSVECTANTS 37 It is easy to see that if the set of variables x^, x^, ... x^ is contragredient to y-^, y^, ... yn\ then 3/1, 3/2, •.. 2/n is contragredient to iTi, x^, ... Xn. For example the symbols a and a; in a symbolical product are contragredient. 39. From these definitions we deduce the following theorem : If x^, X.2, ... Xn; y-i, 2/2 > ••• Vn (^^'0 two contragredient sets of variables, then x^y-i + x^y^ + . . . + ^n1/n is unaltered by any linear transformation. Conversely, if ^i2/i "t" ^2^/2 + . . . + ^n^n is unaltered by any linear transformation, the two sets of variables x-i, X2, ... Xn', 3/1, 2/2, ••• yn cc'^s contragredieut. Let the two sets of variables be contragredient, then if (I) and (III) be the equations of transformation, a-'i 3/1 +^23/2 "r ••• -^ ^nyn — ■Vi {U,\ -3^1 + ^1,2 ^2+ ••• +^l,7l-X',i) + 3/2 (^2,1 ^\ + ^2,2 -3^2 + • • • + 4,n ^n) + I Vn ('»i,i -^1 + 'n,2 -^2 '^" • •• "^ ''n,n ■^n) = ^1 (^1,1 3/1 + ^2,1 3/2 + ... + ln,xyn) + ^2 {li,2 Vi + 4,2 3/2 + • • • + ^n,2 Vn) + -\- 2Ln ('1, 71 3/1 + ^2,w 3/2 + ••• + ln,nyn) = Xi7i+X2F2+... + Z„F„. Q.ED. Conversely let «i3/i+ ^23/2+ ... +Xnyn be invariantive, then if «i, x^, ... Xn are transformed by equations (I) Xi Y^ + X2 Y.2+ ... + Xn Yn = x^yi + ^23/2 + . . . + x^yn = 3/1 (4,1 ^1 + 4,2 X^+ ... + 4,n -^n) + 3/2 (4,1 X^ + 4,2 X2 4- . . . + 4,n Xn) + + 2^»(^n,l -X"! + ^^^ 2 -X'2 + . . . +^„_„X„). r^w 6y9(>3 38 THE ALGEBRA OF INVARIANTS [CH. Ill But this is an identity true for all values of Xi, X^, ... Xn\ hence the coefficients of Xj, X^, ... Xn on the two sides of the identity are equal ; i.e. Yl = ^1,1 2/1 + ^2,1 3/2 + • • • + ^n,l Vn Y2 = 11,2^1 + 12,2^2+ •-■+ ln,2yn ^n — ^1, ra Vl + %m 2/2 + • • • + ^re,n Vn- This set of equations is the same as (III), and hence the two sets of variables are contragredient. Q. E. D. It may happen that a set of variables x^, x^, ... Xn is subject to a restricted group of linear transformations, and that Xiyi + ^2y2+ ••• +Xnyn is an invariant for all transformations which are allowed. The above proof still holds that y^, y^, ... 2/„ is contragredient to x^, x^, ... Xn. But yi,y-i, •-. yn is subject to a restricted group of transformations. Ex. (i). If the binary quantic become after any linear transformation (-^OJ -^15 -^2* ••• -^ni-^i, -A2)", then (tto, «!, ... ctnl^i, •^2)"=(^o> ^i> ••• ^n$J^n -^'2)" Hence ■^OJ ( I j -^n (2]'^^' *■■' '^'" •*'! > "^l •''2> •••» •*'2 » are two contragredient sets of variables, subject to a restricted group of linear transformations. The linear equations connecting the original with the trans- formed coeflBcients of a binary form may be deduced from this fact (cf. § 16)*. * By means of the group here indicated binary invariants and covariants are brought under Lie's general theory of invariants of continuous groups. If we pass from the variables fj, ^^, ... tm to ^^^ variables fj', fj'* •■• fm' hy means of the transformations, fp'=/p(fl, fs' ■■• imJ <*l' «2» ■•• «r)» and these transformations form a group when the parameters a vary, then a function 39-40] POLARS AND TRANSVECTANTS 39 Ex. (ii). Two sets of variables both contragredient to a third set are cogredient with one another. Ex. (iii). If the determinant of transformation of a set of variables be /x, the determinant of transformation of the contragredient set is - . 40. If F(ao, tti, a.2, ... a;^, x^) be any covariant of a binary quantic (uq, a^, a^, ... a^^x^, x^''^ and if y-i, y^ be a "pair of variables cogredient with x-^, x^, then y^dx^y^dx^ is unchanged by any linear transformation of the variables, except that it is multiplied by the same power of the determinant of transformation as that by which F is multiplied after trans- formation. For if F{Aa, A^, A„, ... Xi, X^) = fiF{ao, a^, a^, ... x^, x.^ where A^, A^, A^, ..., X^, X^ are the transformed coefficients and variables, and if z■^, z^ be any pair of variables cogredient with x-^, x^ which become Z^, Zo after transformation, then ^(^0, ^1, ••• Z„ Z,) = fMF(ao, ai, ... z„ z,), fi being a power of the determinant of transformation. But Xi + Xyi, X2 + \y2, \ being any constant, are a pair or variables cogredient with x\, x^, hence F{Ao,Ai,...,X^ + \Y,,X. + XV.^ = fiF(aQ, ai,...,Xi + Xy^, x^ + Xy^). Now F being a rational integral algebraic function of all its variables, we may expand each side of the above equation in powers of X, ; since A. is arbitrary, the coefficients of the different is said to be an invariant of the group if ^(fi'. fa'. •■•fm') = -P'(fl.f2.•••U• In our case the variables are (n + 3) in number, viz. a„, aj, ... a„, Xj, acj, and the transforming equations are all linear. The parameters are the coefficients in the original transformation of .Tj, x^, and if they be so chosen that Iii7.2~^2'?i = l- ^^^^ an invariant or covariant of the binary form is an invariant of the above group in (n+3) variables. The reader will find it interesting to write down the actual transformations for the a's and thence to verify that they form a group. Cf. Lie, Vorlenurifien ilber contiimierliche Gruppen, p. 718 etc. 40 THE ALGEBRA OF INVARIANTS [CH. Ill powers of \ must be the same on the two sides of the equation. Hence, using Taylor's theorem : F{Ao, ^1, ..., Zi, X2) = fMF(ao, tti, ..., Xi,x^) ( ^i^Y + ■^^ov^ ) -^(-^O) -4i, ..., Xi, X2) from which we see that f 3/1 x-;- + 3/2 ^— ) -^ is invariantive, if F is \ OOl^i OtX/2 1 a covariant. 41. The definition of covariants may now be extended thus : Any function of the coefficients of a single quantic, or of a simultaneous system of quantics, and of sets of variables, — all sets being cogredient with the variables of the quantics, — which is such that, when any linear transformation is made and the original coefficients and variables replaced by the transformed coefficients and variables, it is unaltered except for a factor de- pending on the coefficients of transformation. Let a^i, 5^2 ; 2/1, 2/2 be two cogredient sets of variables, then {x^y^-x,,y:) = {xy) is unaltered, except for a factor — which is the modulus itself — by any transformation. Hence, if it be necessary to consider covariants with sets of cogredient variables, we may replace all the sets of variables but one by the coefficients of linear forms added to the system. For we may regard {xy) as a linear form — since it becomes (ZF) by transformation — and hence replace the variables y^, —y^ by its coefficients. Hence covariants having more than one set of variables may be represented symbolically as a sum of products of symbolical factors of the types (a/S), ««, Uy, {xy), — 42. Convolution*. The word convolution is used as a name for the process of obtaining from a given symbolical product (representing a covariant) another symbolical product * German Faltung. 40-43] POLARS AND TRANSVECTANTS 41 (representing another covariant) by removing two of its factors of the form cLx^x and replacing them by a single factor of the form (ayS). Any symbolical product P' obtained from the product P by means of this process either once or several times repeated, is said to be obtained from P by convolution. Thus the covariant {ahf{ac)aJ)^Cx of the quartic is obtained by convolution from the covariant {ahfa^b^Cx*; the factor a^Cx being replaced by {ac). It may also be obtained by convolution from {ah){ac)axhxCx', also from {ah)axhxCx*, or from axhxCx- This process, it should be noticed, is purely a symbolical process, and has no analogue in the non-symbolical treatment of modern algebra. 43. Polars. The operator has already been introduced. In § 15 we defined the form ^'(^a^)> (^-) to be the rth polar of P ; P being a binary form of order n. Again in § 40 of this chapter it was proved that if P is a covariant and the variables 3/1, y^ are cogredient with x-^, x^, then the form (IV) is also a covariant. For the purpose of calculating polars, we may use a theorem identical with that of Leibnitz for ordinary differentiation. Thus if the ?'th polar of a^h^ be required, it may be obtained by operating on this expression with (m + »)! where Pj = y^ ^ — + 2/2 ^ , but operates only on ax'* ; and Pg is the same expression but operates only on bx^. Ex. (i). Find the .second polar of (ab)- ajbj. Ex. (ii). Find the rth polar of a^'^h^^cjj'. 42 THE ALGEBRA OF INVARIANTS [CH. Ill 44. In view of the fact that covariants will generally be given in terms of symbolical letters which refer to the original quantic or quantics, and that in this case the factors a^ are not all the same, we shall consider the general case in which these factors are all different. Results, proved for this case, may be obtained for any other by simply equating two or more letters. Consider now the form where P is a product of symbolical factors not containing x-^, x^. The first polar is " fl V ' ' X + Is n •••««-!, "nj the term in the bracket being simply an abbreviation for We notice in this expression that : (i) The difference between two terms L «r -F'^.oi,^ y " - ] L «r a« J where X is an expression obtained from F by convolution. (ii) The difference between the whole polar and one of its terms F''.ai^-^ i^*".a. POLARS AND TRANSVECTANTS 43 44-45] where X is a sum of terms each obtained from F by convolution, each such term being possibly multiplied by a constant. Similarly the rth polar of F^^ is fli otg . • . Or F. a, cCi (V), where the summation extends to all possible sets of r factors taken from a.-^ a-i ... a« . To see the truth of this suppose it true for the rth polar, then the r + 1th polar may be obtained from the 7'th by operating with — • (y^)- ^o the left this gives n — rX' dxj ■p^n-r-i p^r+i^ Consider one term on the right, that expressed above ; by polarizing we obtain «! tta ... <Xr di ' r\{n-r -1)1 ^^ F^. «! Oa Clr °li Each term of (V) gives rise to a similar expression. Now any particular term of the r + lth polar, arises r + 1 times, once from each of r + 1 terms of the rth polar, thus " tti a^ ...Or Or+i " J/T n y y » » arises from each of those terms of the rth polar, obtained by omitting one of the factors in the numerator and the corresponding factor of the denominator of this expression. Hence ai fla • • • O^r+i ^ , y y _y aj^Og^ ... a,.+i^ Now it has been seen that this is the correct form for the first polar ; hence it is the correct form for the second, and so on ; the formula is therefore true in general. 45. The number of different terms ^ «! "2 • • • Or ^ FJ^ ' " -' pn-r-. p ,+, ^ (r+l)!(»-r-l)! ^ " n ! F^ 18 IS The coefficient of each term as it appears in the polar F^^'-^F/ 1 0' 44 THE ALGEBRA OF INVARIANTS [CH. Ill hence the sum of the coefficients of all the terms of a polar is unity. This remains true if some of the letters a become equal, for no term of the polar can vanish. 46. Two terms of the rth polar are said to be adjacent, when thev differ only in that one has a factor of the form a^ aj, while in the other this factor is replaced by a^ a* • We shall now prove that : (i) The difference between any ttvo terms of the rth polar of F^ is equal to {xy) X, where X is a sum of terms each of which is a term of the r — 1th polar of an expression obtained from F^^ by convolution. (ii) The difference between the rth polar of F^ and any one of its terms is equal to (xy)X, where X is a sum of terms each of which is a term (multiplied by some constant) of the r — 1th polar of an expression obtained from, F^ by convolution. The difference between any two adjacent terms %\ F, - ccha^^ F, = (xy) (a^afc) F^ = (xy) X " ' , r i?!," ~ . where X is a term of the ?•— 1th polar of (a^ajk) ^ ~ , *'-6- a term of the r— 1th polar of an expression obtained from F^ by convolution. Now any term of the rth polar of F^'' may be obtained from any other by means of a finite number of interchanges of letters such as ail, aj^. Hence between any two terms Ti, T^ of the rth polar a series of terms Tj^,, Tj^j, ... T^^i may be placed such that each term of the series -^1) -^1, i> -^1,2) ••• -'i, I) ■'a is adjacent to that on either side of it. Hence the difference between two terms T,-T,= {T,-T,,,) + (T,,,-T,,.} + (T,,,-T,,,)+ ... +(T,,i-T,) = (xy)X where X is a sum of terms each of which jis a term of the r — 1th polar of an expression obtained fiom F^^ by convolution. 45-47] POLARS AND TRANSVECTANTS 45 Again, the difference between the complete polar and any single term T Hr) = 2 i = l T, -T Ti representing the general term in the polar 'T,— f Hr) S and hence the theorem (ii) follows at once by (i). It should be observed that if two or more factors are now made identically equal, the above proof is not affected ; we may for the sake of argument suppose them all different, and make them equal in the final result. The only effect of the equality of factors is that some of the terms obtained by convolution from Fx^ will vanish. The propositions are true, then, as stated, for any symbolical product ; and consequently for any covariant form. 47. Let T be any term of the rth polar of F^^, then by proposition (ii) of the last paragraph where ^^_i is a term of the 5 — 1th polar of a form obtained from Fx^ by convolution and \y_i is nurnerical. Let the r— 1th polar, of which (jjr-i is a term, be -v|r,._i. Then applying (ii) again we have (f)r-i — ^{rr-i = i^y) 2 Xr-2 4>r-2 where ^r-2 is a term of the r — 2th polar of a form obtained by convolution from F^^. Proceeding thus we see that where ■yjr^. is the Jcth. polar of a form obtained from F^^ by con- volution ; and A^ is numerical. 46 THE ALGEBRA OF INVARIANTS [CH. Ill The terms of this series, the existence of which has just been demonstrated, will be accurately determined later. The series is known as Gordan's series. 48. Transvectants*. If aa;"*, 63:'' be any two binary quantics, the form {aby a/"-*- 6/-*- is called their rth transvectant, or their transvectant of index r. The symbol (/ <f>r is used to denote the rth transvectant of two forms/, <f). Thus (a/'^, 6/)'- = {aby a^-^ b^\ The definition of a transvectant just given is symbolical : the process of forming a transvectant is however not a purely symbolical one, — like that of convolution. In order that we may be able to obtain transvectants of any two forms, we make use of the diffe- rential operator, introduced in Chapter I.: n = -^^ ^. Thus fn, t M ! " {m — ry. {n — ry. ' ^ Hence if f{x\ (f> {x) be any two forms of orders m, n re- spectively, then the rth transvectant of / and (f> may be obtained by operating with ^ 1 -^ — r- ' Cl^ on f{x).<f>{y) and after operation replacing y by x. Thus {m — ry. (w — r)! {f{x), <!> {X)y = ^^^^- . ^^^- [nrf{x) . (/> (2/)],., . Ex. For the cubic the first and second transvectants of the Hessian with the cubic itself are {{ahf a^b^, c^sy=^ {ahf {ac) Kc^^-^\ {abf {he) a^c,\ {{ahf ax hx , c^^f = {ahf {ac) {he) Cx ; as may be seen by using the differential operator. * German Vberschiebung. 47-49] POLARS AND TRANSVECTANTS 47 It is useful in calculating transvectants to notice that, just as in the case of polars, the sum of the coefficients of the various terms of a transvectant is unity. 49. For the purpose of calculating transvectants the following method is extremely useful. Consider the rth transvectant of two forms aj^, h^\ it is {ahy a^"*-*" h^-^. It may be obtained by the following rule : Polarize a^^ r times with respect to y, we obtain a^^'"^ aj \ then replace y-^ by h^, y^ by — 6j and multiply by ho^~^, the result is {ahy »«:"'"'■ &»''"'■ which is the rth transvectant of a^ and h^. We proceed to illustrate the method : (i) Consider the second transvectant of the Hessian of a quartic with the quartic itself, The second polar of (ahy a^b^^ is ^ (aby ay^bj^ + | (aby ajy^ayby + ^ {aby aj^by^ Hence the transvectant required = 1 {aby (acy b^'cj" + 1 (aby (ac) (be) a^b^c^^ + J {aby {bey a^d = \ {aby {acy bj'c^' + | {aby {ac) {be) a^b^c^^ since a and b are equivalent symbols. (ii) The third transvectant of these two forms is {aby{acy{bc)b^c^. (iii) To obtain the second transvectant of the Hessian of the quartic with itself: i.e. {{abya,'b,\{edye^'d^J. Let us write {cdy c^di = hx*, where the symbolical letter h refers to the coefficients of the Hessian considered as a separate binary form. Then as in (i) {{abya^'b^Mi^J = J {aby {ahy bJ'K^ + 1 {aby {ah) {bh) aJ)^K\ To obtain the first term we polarize h^* twice, replace y^ by — a^ 48 THE ALGEBRA OF INVARIANTS [CH. Ill and 2/2 by + aj and then multiply the result by ^ (aby bx'- Thus the second polar of h^* = Hcdy Cy'dJ' + 1 (cdy CydyC^d^ + J (cdf C^Hy\ Hence \{aby{aKfbx'K' = jg {ahf {acf (cdy b^H^ + \ (aby (ac) (ad) (cdy c^dj)^ ^^^(aby(cdy(adybx'c^ To obtain the second term we polarize h^* once with respect to y and once with respect to z, and then replace y hy a and z by b. The polar is ^ (cdy CyCzdi + \ (cdy CydzC^d^ + \ (cdy CgdyCxd^ + ^ (cdy c^dyd^. Hence \ (aby (ah) (bh) a^b^hj" = ^ (aby (cdy (ac) (be) ajy^d^^ + I (aby (cdy (ac) (bd) aj^c^dx + f (aby (cdy (ad) (be) a^bxC^d^ + ^ (aby (cdy (ad) (bd) a^b^d. Hence remembering that all four letters are equivalent, we obtain ((abya^^bx\(cdycHry = \ (aby (acy (cdy b^^d^ + 1 (aby (cdy (ac) (ad) c^dM + f (aby (cdy (ac) (bd) aJ)xCxdx. (iv) Calculate (v) Transvectants of the following form are of frequent occurrence : = 2 Uy \m/ ^^^a ^j^ n-Xj^m-^ p-r The result may be obtained at once by polarization, (vi) From this may be deduced the value of Let Cj5Pc?,9=Aa,P + «. 49-50] Then from (v), POLARS AND TRANSVECTANTS 49 T= 2 <T-Vr=r CKI) {aKf {hhy a/- " h^"^-^ h^P+^-''. But polarizing Ax^^* f times with respect to y and r times with respect to z we have ^ ^ X!.!(^-X-.)! ,!z.!(g-^-.r)! , cr\ t! (^ + g'-o- — r)! Hence replacing the y's by a's and the /s by 6's P! r = 2 ?•! (n-X-^)! (m-i/-tzr)! (;?-X-i/)! (y-/x-ar)! {'m + n — r)\' {p + q — r)\ (vii) If /=ai a2^...a^ , 0=^, ^2,... /3„ , (/, <^r= 1 sf (ai/3i) (02/32) -K^r) ^ ,1 where the 2 extends to all possible arrangements of the letters aj, 02, ...a„; ^i,)3„..A. §44. 50. Two important theorems relating to the difference between terms of a transvectant, must now be proved. They are exactly analogous to those already obtained for polars in § 46. (i) The difference between any two terms of a transvectant is equal to a sum of terms each of which is a term of a lower trans- vectant of forms obtained by convolution from the original forms. (ii) The difference between the whole transvectant and any one of its terms is equal to a sum of terms each of which is a term of a lower transvectant of forms obtained by convolution from the original forms. Here, as in the case of polars, we introduce the idea of adjacent terms. Two terms of a transvectant are said to be adjacent when they differ merely in the arrangement of the letters in a pair of symbolical factors. Two terms can be adjacent in any one of the following ways : (i) P (ai^j) (a^^k) and P (a,/8,) (a^^j), (ii) P (ai^j) tth^ and P (ah^j) \, (iii) P (ai^j) ^[ and P (a^/S,) y8-^, G. & Y. 4 50 THE ALGEBRA OF INVARIANTS [CH. Ill where the letters «!, otg, ... belong to the first of the two forms in the transvectant, while ^i, ^2,--- belong to the second form. The difference between two adjacent terms is in the three cases seen to be (i) P(«,ar,)(/3,-/3,), (ii) P{aia„)^j^, (iii) P(^,^j)ai^. To fix ideas we shall suppose that the transvectant we are considering is the rth transvectant of and (f) = B.^^^^^^...^n^, where A and B are products of symbolical factors of the type (yB), and all the factors of the type y^ are different. Then (/.<^r= ^ C) where the S extends to all possible arrangements of the letters tti, Oa-.-o^; /3i,^2-..^n,— § 49 (vii). The truth of this statement may also be seen by operating with Xl*" onf{x) (y) ; and remembering that if we write ai = a2= ...=a^ = a, /3i =^2= .•• =yS„ = /8, then (/ (f>y = (a^y a,"^-^/3^''-^. The difference between two adjacent terms of the above trans- vectant is a term in which at least one factor of the type (ayS) is replaced by a factor of the type (oca'), or else of the type {^fi'). There are then not more than (^ — 1) factors of the type (ay9). Hence the difference between two adjacent terms is a term of a transvectant of index less than r, of forms obtained by convolu- tion from / and <j). Thus, for example, (abf {acf h^ c^, (aby (ac) (be) aja^c^ are adjacent terms of the transvectant {{abya^^bi,c^y, and their difference {aby(ac)b^c^^ 50-51] POLARS AND TRANSVECTANTS 51 is a term of {{abfaj)^, Co;*). Now between any two terms Ti, T^ we may place a series of terms -'i, 1> -'i, 2» ••• -^ 1, i such that any term in the series 1> -*^1, 1> -^ 1, 2> ••• -^ 1, 1> -^2 is adjacent to that on either side of it. For we may obtain T^ from Ty by a finite number of inter- changes of pairs of letters, a pair being composed either of two as or else of two yS's ; since each letter occurs only once — in our argument, — the terms which differ by the interchange of a single pair are adjacent. Hence the difference between any two terms i.e. a sum of terms each of which is a term of a transvectant of lower index of forms obtained from the original forms by convolu- tion. This is the first theorem. Again if T be any term of the transvectant, then (/, (jiy-T= — !— - tr - T r) \r. 2 {T - T) [since the number of terms T' is, r\{ j ( j J and this is equal to a linear function of terms of transvectants of lower index of forms obtained by convolution from y and <j>. 51, This theorem may be extended by applying it again to each of the terms of transvectants of lower index on the right- hand side. This may be done repeatedly. Each time the process is applied the terms of transvectants on the right are replaced by the transvectants themselves, and linear functions of terms of transvectants of lower index. After not more than r applications 4—2 52 THE ALGEBRA OF INVARIANTS [CH. Ill of the process, we have on the right a linear function of trans- vectants of forms obtained from f, <f> by convolution, whose index is less than r, and of terms of transvectants of zero index. But a trans vectant of zero index of two forms is simply the product of the forms ; a term of such a transvectant is merely the same product, and therefore the transvectant itself We obtain then the following important theorem : The differ- ence between any transvectant and one of its terms is a linear function of transvectants of lower index of forms obtained from the original forms by convolution. Ex. (i). {abY(bcya^W = {iabf a,^h,\ c,*)2 - % {ahY (be) a,cj^ - J {ahf (ac) b^c^^ = ((a6)2 a^^bj', c^^Y - (Sflbf a^b^, c^*) + § {abf . c^\ Ex. (ii). Prove that {{ab)a,^bj^,{cd)cj^d,^f = ^iab) (cd) {{boy {adf+9 (bcf {adf {ac) (bd) +9 (6c) (ad) {acf {bd)^+{acf {bd)^} = J {(&«)* («^* - («c)* {bdf} - \ {abY {cdf {{bcf {adf - {acf {bdf} = i {Q>cf (ad)* - {acY {bdf) - \ {abf {cdf {{be) {ad) + {ac) {bd)} = J {{be)* {ad)* - {ac)* {bd)*} - 1 ((a6)3 a,b, , {cdf c^d^f. Ex. (iii). If /be a cubic and H'\\& Hessian, prove that {H,ff=0. Ex. (iv). If/ be a quartic and ZTits Hessian, prove that {ff,f)^ = 0. Ex. (v). If bx* be the Hessian of a^*, and {ac)* = = {ad)*, then {{ab)a^^bxMcd)ex^dx^f=0. Ex. (vi). The second transvectant of a quantic / of order n and its Hessian II is where i={f, /)*. For V 2 ) = (a*)2(ac)2 a,'»-46.»-2o.»-2- gj^^ t/ on using (III) § 22. Finally the result is obtained by the help of the last identity of § 22. CHAPTER IV. GORDAN'S SERIES. 52. Gordan's series. In § 47 it was proved that any term T of the 7wth polar of a symbolical product F can be expressed in the form X{xyyFi; where Fi is the (m — i)th polar of a sum of symbolical products — possibly multiplied by positive or negative constants — each of which is obtained from F by convolution. The product a^hy^ is a term of the with polar of ax"})^, hence a,«6,- = 2(a;2/yi^<V-* (I); where the suffix indicates that F'^^^^-i is the (m — i)th polar of F^K This is an identity; it must therefore be true when y = x*\ hence for when y — x, {xy) = 0. Now if in a function of x polarized with respect to y, y is replaced by x, the result is the same as the original expression. Thus the rzth polar of a,^'^ is a^ay^, which becomes a^'^'^ when y is replaced by x. Hence a^H,-^ = \F<\m\^^=F' (II). Thus the first term of the series (I) is obtained, the other terms may be obtained in a similar manner. Operate on (I) with IP'. Consider first the effect of this operation on the term {xy)'F^\,.-i. * This of course means — = — . 54 THE ALGEBRA OF INVARIANTS [CH. IV By comparing the degrees in x on the two sides of (I) it may be seen that the degree of F^^^^-i in ^ is n — i. Hence we may write and F^^ym-i = a^^-'a/'-\ Now = i(m+n-i + l) {xyf-^(i^-^(iy^-\ a result obtained by ordinary differentiation as in §§ 26, 27. Hence by repeated operation of VL we obtain \ y' ^ y {^-J)\ {m + n-i-j Jriyr ^' ^ v ' when i >j; but when i <j nJ {xyfaj'-^oiy'^-^ = 0. As regards the right-hand side of the equation (I) we know, § 27, Ex. (i), that nJaJ'by'^ = ; ~, . 7 ^, {ahya^^-iby^-i. (n-jy. (m-jy. ^ The result of this operation is, then, in-jy.'im-jyr^^'''' ^' = 2 y. — TT^, -. — -. — =-r-, {xyf •? J^'*V-<. i=i (t -;)!(w + w - 1 -J + 1)! ^ ^^ ^ In this put y = x, and we obtain (w-^)! (m-;)!^ * * (m + w-2j + l)! and hence F(3) = ^J^^3/ {abya^^'-^hx'^ in 52-53] gordan's series 56 Substituting this value of ^'■^"' in (I) we have Gordan's series : — ^n\ /m i=o /m + n — i /m + n — i + 1\ 53. This series may be put into other, rather more general, forms. Multiply by {aby, write « + ^ for n, and m + r for m, then «-.--.(M^)«<..-.«;. Operate with ix-^], and we obtain ^n — r\ fra — r — k i fn — r\ fi Operate with [y ^] on (IV), then m—r 'y ^n — 1 — k\ fm — r = 2 +„-2;-.>i\ ("^>'('"" *«-);:-— - (^i)- In (V) replace y^ by Cg, 3/2 by — Ci, and multiply by c^P"^'^^'^^, we obtain — see § 49 — 'n — r\ fm — r— k 7)C = X{-iy " " ' \ — V-^ ((aa;^ bary^'-, c^^)"»-*-^-*...(vii). 56 THE ALGEBRA OF INVARIANTS [CH. IV The equation (VI) gives a similar result when the weight of the covariants under consideration is greater than m the order of fea,*": viz. — (aft)*- (6c)'™-^ (ac)*aa;''-^-*Ca;2'-"'+'-* /n — r — k\ /m — r\ = S ( - 1)^' L + L2r-/+l( ^^^»''^' ^*">'^' c.^)-i-+* . . . (VIII). \ i J 54. Now if in (VII) a and c, n and p, r and m — r — k are interchanged, the left-hand side of the relation is unaltered, except for a factor ( — 1)"^~*. Hence fn — r\ .Vfi — r — k\ fm + n — zr -i+\\ 'p + r-'tk — m\ fr\ = 2 ^ ^ ^ . :, ^-^-frv ((^*"'. 0^^)^+"^'--*, a*")''"*- '^ + 2r-l-2« — w — t + l\ On the supposition that f=h^^, <i> = aar, '^^cJP, a, = 0, a^ = m-r-k, as = r, this relation is the same as % A ! l\ll^((f A,)<H+i y}rY: + <h-i , = ( - 1)"' S ^^ \ .\ (( /, ^|r)-.+», <f,)».+«3-* . . . (IX). ^ ^ /m+p - 202-1 + ly^-^' ^^ ' ^^ ^ ^ Again, if in (VIII) we interchange a and c, n and p, r and m — r respectively, the left-hand side of the relation is unaltered except for a factor (— 1)"»"^. Hence we obtain a relation of the same form as (IX), but in which ai = k, 0Lz = m — r, ct^ = r. 53-55] gordan's series 57 Thus the identity (IX) is true in two cases. In the first ai = : as cannot exceed either m or n, for we use in (VII) the factor {aby : 0^+ cis — m — r and therefore cannot exceed m : and finally a^ cannot exceed p. With these restrictions Oa, a.^ may take any positive integral values. In the second case tta + as = ini- ttj + «! cannot exceed n, and a^ + aj cannot exceed p ; subject to these restrictions a^, a^, a^ may take any positive integral values. The series (IX) is one of great importance for calculating transvectants. It is usually quoted in the form / f <f> f \ { m n p ) , \ «! 02 Ots / «•> + Ofs > W, tts + fli > n, «! + tta > p, where and either (i) fli = or (ii) aa + Qs = wi. 55. In order to illustrate the use of the series, we shall now calculate some transvectants which are covariants of the sexticf*. Let us write H = {f,f)\ i = (f,fy, A=(f,fy, t = {f,H). Then to calculate the transvectant (H, fy we use the series / / / / \ (666), \ 3 2 / (7) (7) * other examples will be found in Chapter V. 68 THE ALGEBRA OF INVARIANTS [CH. IV that is {H,fr+^{i,f)={i,f), or {H,fr = \{i,f). The object usually in view when calculating transvectants, is to express them in terms of other transvectants of lower index. There is no difficulty in choosing the series which will give the desired result, in general we select it so that the transvectant to be calculated appears as the first term on one side of the identity, while all terms on the other side are of lower index. The transvectant (H,/)* is given by / / / / / / / \ (666) \ 4 2 / ((/ /^^ fy-' = ^4P4 ((/ n*^'' /)'-'> hence CT) {'?) iH,fy + ^-^{i,fy+lA.f=(ify + lA.f, or {H,fy = ^(ify + ^A.f. To calculate (H, fy we cannot write ai = 0, for then the condition a^ + aai^m would be violated. We must then make ttg + ctj = m = 6 and use the series / / / // / / \ (666) \ 1 4 2/ hence i ) K i I (ff,/)'=-y*'(i,/r. * This form vanishes identically — see Ex. (ii). 55-57] gordan's series 59 To find (t, fY we use // H f. 16 8 6) \ 2 1 / whence using the result, § 51, Ex. (vi) we obtain Ex. (i). Prove that for the sextic (zr,/)«=^(t,/)*. Ex. (ii). By means of the series / / / 6 6 6 13 3 prove that (i, f)^ = 0. Ex. (iii). Prove that iif,^r',ff=-^QU^r,n {H, ^•)4=2^^•,^•)2 + ^^.^•. 56. The series we have been illustrating does not give all the relations between the covariants of/, ^, -^ which are of weight «! + «2 + "s- For example, we cannot by means of this series reduce the form ((/, fy, fY which is reducible whatever be the order of/, § 51, Ex. (vi). 57. The series (III) may be inverted with the following result n («) (a,™, 6=.»)V = S(- l)*^)£;^(aJ)'a,"'-'6„»-«(^2/)' (X). i I m "T '^^ \ i J (Gordan, Invarianten-theorie, § 7, pp. 89, 90.) 60 THE ALGEBRA OF INVARIANTS [CH. IV To prove this, we first establish the existence of an expansion of the form indicated ; and secondly we find the coefficients. To prove the possibility of such an expansion, we observe that = l\ia^^-%''-^ [aj)y - {ah) {xy)Y = S/Uj {aby {xyy a^-i hy^'-i. To determine the coefficients /u,, we operate on the relation just proved with ft ; the left-hand side of the relation vanishes, and hence by equating the coefficients on the right to zero we obtain yLtii (m + w — *■ + 1) = - /ii_i (m — I + 1) {n — i-\- 1). Replacing y by a?, we see that /io= 1, hence in general f^i = (-) 7 m + ri Other series may be deduced from (X), by the same methods as were used in § 53. Thus f7n — k\ fn — k I / \ I (aoT, h^fyn-u = 2 {-y \^/^_2M («^)''*-*«^""'"* V"'-* (^)^ ((«*"*, hx^^f, cjy 'tti — k\ (r { = S ^ — " ' 'V. (aby+''{bcy-' a^"^«-*&a;"-'-*Ca;^-'-+^' . . .(XI), /m + n-2kY ^ ^ ^ ^ ^ when r if-n — k. fm + n- 2k — 7'\ hi — k\ (K"*. &*")*, cx^r = S -^ r^-r 1 , ' X (a6)»+* (6c)"-*-* (ac )*+»-" a^m+n-2*-r-tc/-r+f (XII). 58. It has already been pointed out, § 41, that the system of concomitants belonging to a binary form with two or more cogredient sets of variables, is the same as that obtained when 57-60] gordan's series 61 certain binary forms with only one set of variables are taken for simultaneous ground-forms. Gordan's series (III) shews us which the ground-forms must be ; thus for the form we consider the system ax^hx, {ah) ax, {xy). Each member of this system is unaltered by linear transforma- tion, and ax^hy can be expressed in terms of the forms given. 59. Ex. (i). Any symbolical product having two cogredient sets of variables x, y, can be expressed in the form where P is a symbolical product containing only one set of variables x. If we put Xy^l, .^2 = 0, yi = 0, ^2 = 1. then Pym-i becomes the coefficient oi x-^~'^'^^x^~^ in P. Hence we may express any rational integral function of the coefficients of a binary form linearly in terms of the coefficients of its covariants. (Elliott, Proc. London Math. Soc. vol. xxxil. p. 213.) Ex. (ii). Express the product ao'^i*2 of the coefficients of the cubic (<^0> '^IJ '^2) ^X-*^!} •^2/ linearly in terms of coefficients of its covariants. 60. If a symbolical product representing a covariant of a single binary form contain a factor (aby^~^, it may be expressed in terms of products each containing a factor (ab)^. Let the order of the binary form be n ; and let P be the covariant in question. Then since P contains the factor (a6)^^~^ it is evidently a term of a transvectant {(abf^-'ax''+'^^bx''+'-^\ (f))", where (f> is some other covariant. Hence by § 51 P = ((a6)'^-iaa,"+i-^6a,"+i-=^\ (f>y + tC {{aby^-'ax''^'-'^ 6^"+^-^, </>)"' (XIII), where C is a constant, and ^ denotes any function obtained from <fi by convolution. 62 THE ALGEBRA OF INVARIANTS [CH. IV Now for h and a are equivalent symbols. Hence all those transvectants in (VIII) in which no convolution has taken place in the first form are zero. But every form obtained by convolution from (a6)^^~^aa;"+^~^^6a;""^^~^^ has a factor {ahf^. Hence every trans- vectant in (VIII) either vanishes or is the transvectant of a form having a factor (ab)^, with another form : and hence P can be expressed as a sum of terms each of which has a factor {aby\ A convenient way of expressing this is to write P = 0, mod {aby\ 61. Owing to the large number of symbolical products which represent covariants.of given degree and order it is important to obtain methods of classifying them. For this purpose the greatest index of any determinant factor in the symbolical product is chosen. We shall use the word grade to denote this index. If the symbolical product is a covariant of a single binary form, then, by the theorem just proved, co variants of odd grade may be expressed in terms of covariants of higher even grade. On this account the German equivalent Stufe is used for half the index. We prefer to define grade as the index itself, since the classifica- tion is useful when the symbolical product is not merely a covariant of a single form. 62. Covariants of degree 3. We proceed to obtain criteria, by means of which it may be at once determined, whether or not a given covariant of degree 3 can be expressed in terms of covariants of higher grade. These were first obtained by Jordan* in 1876. They were independently discovered by Strohf; and his method of investigation is given here. We consider first the covariants of weight w which are linear in the coefficients of each of three binary quantics /i = a,r^ , /2 = K""' , fs = c^"'- Let us write Ui for (be) a^, Wj for (ca) b^, m., for (ab) Cx ; then ^1+^2 + ^3= 0. * Liouville, 2 S6r. in. 1876. t Math. Ann. Bd. xxxi. p. 444 et seq. 60-62] gordan's series 63 Hence v^a-iui = (- Vf-i m/ (w2 + Ms)^"^ ^=" (?) Multiply this expression for u-P~^u^ by where h-^, k^ are any positive integers, and take the sum of each side of the result from i = Q to i = k2: hence = (- 1)^^+^ i^ (^) (^ ~ ^ ~^' ~ ^) t^/--t.3^ t (XIV). Now when g — ky> X > g — ki — k^ — \y therefore the last sum written down may be divided into two series, viz. \ = up to g-k^-k^-l, and \ = 5r - A^i up to ^. Let g — ki — k.2—l=k3. Then the first of these series is (writing i for X) In the second series, write g —i for X, then it becomes lf^)r'-t'~^U"-v (XV). * ( ) is the coefficient of x^ in the expansion of (1 + a;)'. t For S (^T^) ( " \~^) is the coefficient of x*» in (l + ar)^"'*' . (l + x)-^'-l. 64 THE ALGEBRA OF INVARIANTS [CH. IV Now if I and m be positive integers \m) ^ ' m\(l-iy. ^ ^ \ m ) 'l-\-m-V = (-Y , ^ ^ V ^-1 -(-r^'-'{~P_~i^) (XVI). Hence the series (XV) becomes |„<-)^"*-'(f)(t-~i>''""'^' =<->'•"■!„(?) (t- /)'"'"'<"■ +'^>' But the coefficient oiv^^~^Ui^ is here equal to ^ (9\ (9 - ^\ (- h - 1' \,\J \i — X./ V ki — i ^ (9\ (^1 + A^s - \ Hence the series (XV) becomes By means of the relations (XVI), the series on the left-hand side of (XIV) becomes Hence the equation (XIV) may now be written +(->'• I.© f'^fe "')"=""'"''=" <^™>- 62-63] gordan's series 65 This equation is true for all positive integral values of k^, k.,, k.^ for which k^-\-ko,-\-hi = g - 1. 63. Let us suppose that w, the weight of the covariants under consideration, is not greater than the order of any of the three quantics. Then any one of the covariants may be expressed in terms of the set where t = 0, 1 , ... w. For since n.^ -^ w, whenever a factor (ca) appears in a covariant of this weight, it may be removed by means of the identity (ca) hx = — {he) ax — {ab) c^. Similarly all the covariants may be expressed linearly in tei-ms of the set (6c)"'-' {cay a/''-* hx''^-^'-'^ c/''""', or of the set (ca)"' - '■ {ahy a^'^ - *" h^'^- ' » c/3+« - w_ We shall suppose the members of each of these sets arranged in order according to increasing values of i. Then the forms of any one set are linearly independent. For suppose a relation to exist between the forms of the first of the above sets. Let the terms in this relation be arranged in the order indicated. The relation still remains true if we take for fy,f 2, fa special quantics instead of general ones. We shall suppose them to be merely powers of linear forms ; the result of this is that the letters a, b, c are no longer purely symbolical. Hence if Xi {ab)""-' {bcY aa,«'+^-«' i^;"^-"' c^,-"'- ' be the first term, we may divide the identity by {bey. Every term of the quotient except the first contains {be) as a factor; but the quotient must be zero, hence the first term must vanish, when we make b = c, ie. (aby-'-' aa;'*'+'--«' ft^w^.+ws-w'-' = which is clearly untrue. Hence no linear relation can exist between the forms of one set. And therefore there are exactly w +1 linearly independent covariants, which are of the first degree in the co- efficients of each of three quantics, and of weight w (:f> 7ii, Tig or n^). G. & Y. 5 66 THE ALGEBRA OF INVARIANTS [CH. IV 64. Writing in (XVII), for Ui, u^, u^ their values in terms of a, b, c, we obtain a linear relation between the first ki + 1 members of the first set, the first A^a + 1 members of the second set, and the first ks+l members of the third set, where ki + k2 + k3 = g — I =tu — 1. Thus we obtain a relation between w+2 forms. Hence if we take the first irii forms of the first set, the first niz of the second, and the first n^ of the third, where nil + rrii + nis = w + 1, we have a set in terms of which all other of these covariants can be expressed. This set may be chosen so that it contains 2w no covariant of grade less than -^ . For if w = 3m — 1, we may take and we see that all covariants can be expressed in terms of those whose grade is 2m at least. If w=Zm - 2, we may take and we may express all covariants in terms of those whose grade is 2?h - 1 at least. If w=3m. we take mi = m + l, m.2 = 7n^ = m and we have to include one covariant of grade 2m, the rest being of grade 2m +1 at least. In the second case there is one relation between the 3m covariants whose grade is <|;2m— 1 ; viz. by (XVII), (a6)2»» - 1 (6c)"* - 1 aj;»i - 2»* + 1 b^'h - 3m + 2 ^^..3 - m + 1 4- (6c)2"» - 1 (ca)"* - 1 a A " »» + ' 6a;"* ~ "'"" * '■ t'x"' - 3"' + 2 + (oa)2»» - 1 (a6)'»~J aj:"i -3m + 2 ft^Tiu - m + 1 g^H, - 2», + 1 = 2C^ , (XVIII), where C'2,,, denotes a covariant whose grade is <t:2»!. In the third case there are two relations between the covariants of grade ■^2w ; these will be found from (XVII) to express that the difference between any two covariants of grade 2tn and weight Sm, 64-66] gordan's series 67 It would be sufficient for the general theory to prove the theorem : — If Uj + it, + Us = then there are w + 1 linearly in- dependent products of u^, U2, u-s of order w such that each contains an exponent of — at least : this is not easy. 65. Next let lu be greater than the order of one or more of the quantics. If iv>ni, the sum of the indices of (ab) and (ac) cannot be greater than n^, and hence (6c) must have an index equal to w — III at least. We define the quantity e,: to be w — Ui if w > ni, and to be zero in the contrary case. Then each of our covariants of weight w must have a symbolical factor {bcy^ {cay^ {aby\ The remaining factor will be a symbolical product representing a CO variant of weight W — €1 — e.) — €3 = •57, of the quantics ^^M,-e.,-e3^ J^Mj-ea-fi^ p^^Ms-n-fj. But the weight or is not greater than the order of any of the three quantics : hence we may apply the results of the last paragraph. That is, all the covariants of weight w may be expressed in terms of ct + 1 of them. If ei, e.^, €3 are unequal our choice of covariants in terms of which the rest are to be expressed may not be the same as before. But the student will have no difficulty in writing down the result as regards grade. Ex. Prove that any covariant of degree three can be expressed in terms of covariants of grade — .^ — at least. 66. We have now two different series which may be used for obtaining relations between covariants involving three symbols. These are series (IX) due to Gordan, and Stroh's series (XVII) ( - 1)*-^ % rj) (^' '^^' ~ *) (ab)^ - '■ (bey a^''' - «>+' 6/^ - '" c/^ ->...= 0, where ki + A'a -f- A^s = w — 1 and w is not greater than any one of the numbers r'l, ?'2, n.,. If this condition as regards lo be not satisfied we write w — til = 61, w — n., — e.i, w — ??3 = fj 5—2 68 THE ALGEBRA OF INVARIANTS [CH. IV where it is understood that 6 = if n>w. Then the reduced weight ta- is W — €i — 62 — 63. In this case Stroh's series is obtained from (XVII) by writing ■53- for w, Wi — €« — 63 for )h, n^ — ea-ei for n.^, ns-Ci — e^ f'^r "s, and multiplying the result by (6c)'i (ca.y^ (abyK Here k^, k^, k-s satisfy the relation ^'1 + A^a + k^ = 'S7 — 1. The advantage of Stroh's series is that it gives all possible relations between the covariants under discussion. It is, however, generally more convenient to have relations between transvectants than to have them between symbolical products. Thus although series (IX) does not give all possible relations, yet it is frequently the more convenient one to use. By means of series (VII) and (VIII) we may translate Stroh's series into a relation between transvectants. In fact this is what Stroh himself does. This relation has the disadvantage that the coefficients in it are themselves series. It is convenient to have a short method of referring to Stroh's series ; we therefore introduce the scheme used by Stroh /i 72 fa tti rCft fC^ which is distinguishable from Gordan's scheme by the weight being indicated outside the bracket. 67. The quantics of low order furnish very few examples of covariants containing three symbols concerning which Gordan's series gives incomplete information. We have mentioned (f, H)^ as one such case. The co variant ((/, if, /)'' of the sextic / — where i = (/,/)* — is another. The reader will have no difficulty in proving that Ex. Prove that the covariant of the binary form ax^^ = bx^^ = Cx^'' 66-69] gordan's series 69 can be expressed in terms of covariants of higher grade ; and that the covariant {ahf {acf (bcf W^ c^^ vanishes identically. 68. Let the three quantics /i,f-2,f3 be made identical, so that Then if lu ::j> n, all covariants of weight w and degree n can be expressed linearly in terms of those whose grade is not less than 2w 3 ■ If lu = 3w — 2, it is possible to go a step further ; we may express all these covariants in terms of such as are of grade 2m at least ; since covariants of a single quantic of odd grade may be expressed in terms of covariants of higher even grade. Hence if w = Sm— 8:if'-n, 8<3, all covariants of degree three and weight w can be expressed linearly in terms of those whose grade is 2m at least. Similarly if w> n, we have ei = €-2 = €s = € = w — n, ■S7 = w — 'Se. Then we may express all covariants of degree 3 and weight w in terms of those whose grade is not less than —- + €=-- — e. o o If € is odd the lowest grade given by this is odd unless ct = Sm — 2, in which case we see on multiplying (XVIII) by (aby (bey (cay that we may express covariants of grade 2m— 1, in terms of covariants of higher grade. Hence if e is odd, all these covariants 2w may be expressed in terms of those whose grade -^ — — e + 1. If e is even, then as before we take tv = 3m — S, 8<S, and the minimum grade becomes 2m — e. 69. There is one further point in this matter to be noticed. It has been proved that all covariants which are linear in the coefficients of each of three given quantics and are of given 70 THE ALGEBRA OF INVARIANTS [CH. IV weight w can be expressed in terms of those whose grade is higher than a certain number. Further it has been shewn that among the covariants actually retained, no linear relations can exist. We passed to covaiiants of a single quantic, and deduced that all covariants of degree 3 and weight iv can be expressed in terms of those whose grade is higher than a given number. Is it possible that amongst the covariants retained here, there may exist other relations which do not appear in the general case ? To answer this question we express the symbolical products as transvectants by means of § 53. The product (ab)'^ - *■ (be)' ax"'+' " '" bx''^ ~ '" Ca;"» " ' where as before C,„ indicates a covariant whose grade is not less than m. The covariants retained will be — when w i^ n — {(fuAYjsr, {(AfsYj.f, ((/3,/)»,/.y where a ^j; -^ . If ?(; = 37/i — 2, there will be one relation : and if o w = Sm there will be two relations between these covariants. Let us suppose all the quantics to become identical. Then those covariants for which a is odd vanish. Let us suppose 2'W that amongst the remaining covariants for which a-^ -q- a relation exists, say Then using Aronhold's operators which may be written for short (/ ^j , (/2 ^j , (f, ^j we obtain — since Oi is supposed even. Hence corresponding to a relation between the covariants for a single quantic, we may deduce a relation between the corre- sponding covariants of three different quantics. The results obtained then, for a single quantic, are as complete as those from which they were deduced ; and no linear relation can exist between the retained covaiiants. 69-72] gordan's series 71 70. The covariant («6)^ (bcY (cay aa;''-"-^ b^'''^-'' Cx"~''~'' of the binary form a^^ = b^^ = Cx\ can be expressed in terms of n covariants whose grade is greater than \, provided that A- ^ ^ and IM + v> -^; unless \ = ^ = 1/ = - . The verification of the above important theorem is left to the reader; it is really only a restatement of the theorem of §68. It should be noticed also that a similar theorem is true when the letters a, b, c do not refer to the same quantic ; it is {aby- (bey (cay aa;'*'-"-^ bx'"'"^''' c^^-*"-" = S C^+i, provided that fx + v>^. 71. Ex. Prove that the following covariants of f=aj^ vanish identically : ((/, ff\ ff " ^ when n = 4A, 4X - 1, or 4X - 2, {{fjf\fT when n = 4X + l, {{fjf^^'\fT~'^ when 7i = 4A + 2. And shew that no other covariants of degree three vanish except those included in the form U.ff''^\ff- Stroh. 72. Covariants of degree four. It is the object of the present paragraph to determine the conditions that a covariant of degree four and grade \ ( where X,:|> - J may be expressible in the form n n n 2C;,+, + (a6)2(6c)'2(ca)2.^: the expression C^+i denotes — as before — a covariant of grade not less than X, + 1, and ^ being a covariant of degree one can only be the quantic itself. Of course the second term cannot appear if n is odd. 72 THE ALGEBRA OF INVARIANTS [CH. IV It has already been proved that any covariant C = {ahy (bcY (cay a^''-"-^ b^''-^-'^ Ca;"~''~^ where fM + v>-^, is of grade gi-eater than \, unless Auy covariant obtained by convolution from G is of the same form as C, but the indices of some of the determinant factors are increased ; hence any covariant obtained by convolution from G either is of grade greater than X., or else is the invariant n n n (abf {bcf (caf . Any transvectant (G, Fy, where F is any binary form, is also of grade greater than X or else, if p = and \ = fj, = v = -^, has a factor n n n (a6)2(6c)2(ca)2. Further, any term of this transvectant differs from the whole transvectant, by a linear function of transvectants {c,Fy', where C and F are obtained by convolution from G and F. Hence any term of a transvectant {G, Fy is equal to lG,+, + (abf^(bcy(ca)K^, where as before G^^+i is used to denote a covariant of grade X + 1 at least. Again, any covariant having a symbolical factor (abY (bcY (cay, where /x, + y > ^ , is a term of a transvectant (G, Fy, and hence may be expressed in the form SC;,+i + (a6)2(6c)2(ca)2.^. Consider now the covariant of degree four where v If-X K = (abf (bey (cdy aa,""^ bx''-^-" c^"-''-'' d^'"". gordan's series 73 If /i > - , then K= tC^+, + {ahf {hcf {caf . ^. Otherwise by means of the relation {cd) ax = (ad) Cx — (ac) dx we obtain, since ^ <t: X <^ i^, ^ = 2(-)<(;)£„ . where Li = (abf (bey (ac)'^^ {adf aa,"-^-" hx''-^-'^ c^'-i'-'^^ dx"^. If either . \ or . \ then Li = SC,+, + (ab)hbc)hca)i ^. But if yu. + 1/ > X, one of the above inequalities must be satisfied ; hence in this case n n n K = 20x+i + {abf (bey (caf . ^. Next consider the most general symbolical product ^ of degree four ; it is sufficient to write down its determinant factors, which we take to be (aby (acY' (bcY' (ad)"' (bd)"^ (cd)"*. It is supposed here that no index is greater than X, which is n itself not greater than ^ . By means of the identities (cd) ax = (ad) Cx — (oa)) dx, (cd) bx = (bd) Cx — (be) dx, we can express K in terms of covariants L in which either (i) the index of (cd) is zero, or (ii) the indices of both ax and bx are zero. 74 THE ALGEBRA OF INVARIANTS [CH. IV The indices for the covariants L will be denoted by accented letters. In the second case X-\- fj^ -\-Vi =11, \ + fi^' + V2' = n, hence fh' + fjh+ vi + v^ = 2?? - 2\ ^ 2X, consequently either /a/ + /a./ > 2 or i/j' + j/g' > 2 • Therefore K=XL= SOx+i + {ahY {hcf {cay ^. In the first case A^l' + tl2 + Vi + I// = /tti + /*2 + ^1 + ^2 + J'S, and if this sum is greater than \, the same result is true. Hence — Wken fiy + n^-v Vi-\- v^-\- Vz> \ the covariant of degree four ivhose determinant factors are (abY (acY^ (bcY' (adY' (bdY' (cdY\ may be expressed in the form XG,+, + {aby(bcf{cay^. 73. Any covariant which contains the symbolical factor (abYibcYXcdY, n where /a + j/ > \, and X:lf> ^, may be expressed in the form tG^+. + iabf'ibcf^cay'^. For if r be such a covariant, and K = {abY (bcY (cdY «x''~^ &/"^"'' c,^"-''-'' dj'-" ; then _ _ Each term of this sum has just been proved to be of the required form. 72-76] gordan's series 75 74. Any covariant of the covariant Ar^2n-2>- = ^ahy a^"-'- 6^"-'-, (7- :|> |^ of the binary qumitic f= Ux^ = bx' = ..., may be expressed in the form n n n tC,+, + (ab)HbcY(cay^, where Cr+\ represents a covariant of grade not less than r-\-\. To prove this we observe that any form obtained by convolution from a product {ahy ax''-'' bx''-'' {cdf Cx''-' dx""-"" ■ . . is either of grade greater than r or else has a factor of the form (aby{bcy(cdy. In the latter case by § 73 this covariant may be expressed in the form 71 n V ^Cr+^ + {aby'(hcy{cay.^. Now any covariant of kx^"~'-^'' may he expressed in terms of transvectants of the form ({k, ky, n. But the transvectant (k, ky is a linear function of covariants of / obtained by convolution from {aby ax''-' bx""-'' . {cdy Cx''-'' dJ'-\ Hence the theorem is true for covariants of the second degree ; and therefore for all covariants of kx^'"^''. 75. It is well to notice that nowhere in the last three paragraphs has it been assumed that two different symbolical letters refer to the same quantic. The theorems are thus true when some of the letters refer to different quantics. For example the theorem of § 74 is true for covariants of lgm-'2r ^ (^jy ^^n-r J^^n-r when the quantics ax', bx" are different. 76. The discussion of covariants of degree four ma}' be carried on a step further. Thus if X is even, and fi + p=\:^-, 76 THE ALGEBRA OF INVARIANTS [CH. IV then the covariant K= {ahf {bcf (cd)" a^"- Hj''^-" cj' " '^ " " tf^" " " differs from 2, by an expression of the form To prove this we notice that the index of 6^ in ^ is » - X - /x, and this is not less than v ; for by the inequalities above we see that hence we may use the identity (cd) hx = {bd) Cx - {be) dx to obtain A--2(-1).Qa where Now if . X • ^ or ii + v-i>-, then A=SC\+i. (The weight of ^ is 2X ^« ; hence no term with the factor n n n {ab)^{bc)^{cayi can appear.) Then the only term we need consider is X A _^ _6 L, = {abf{bcfibdfaJ'-'bJ'-'^C''d^'''\ 2 Again, let K' be that particular covariant of the form K for which ^=0,v=\. Then 76-77] gordan's series 77 Hence In just the same way it may be proved that if X is odd, and (1) f. + v=\:^-, then K=2C\^^ : n or (ii) /x + v + l=X j>-, then X- 1 X-1 2 The results of this paragraph may be stated as follows : (i) IfXbe even and fi + v = X, then the covariant {aby {bcY (cdy a.,."-^ 6^;"-^-'" c,."-'^-'' c?.,."-" can be expressed as a sum of a reducible covariant and covanants of grade greater than \. (ii) If \ be odd and fi+ i' = X, then the above covariant can be expressed as a sum of covariants of grade greater than \. IfXbe odd and /ji + v = X—'l, the covariant can be expressed as a sum of a reducible covariant and covariants of grade greater than X. Ex. (i). Any covariant which contains the factor {abf {acp {bc^' {ady (bd)"' {cdY\ where ^i+/x2 + j'i + i'v, + i'3>X and X<-, can be expressed linearly in terms of covariants whose grade is greater than X. Ex. (ii). Prove that if a covariant C of degree 5 has the factor written down in the last question, for which /ii + /H2 + «'i + i'2 + i'3=X, and X:f»-, then C='2,C^,^+ reducible terms. 77. Jacobians. The first transvectant of two binary forms fx^> 4^x^ is called their Jacobian. It is, in fact, equal to mn d {Xi, Xj.) 78 THE ALGEBRA OF INVARIANTS [CH. IV The following properties of Jacobians are important. (i) If f, (f>, y^ he three binary forms, each of order greater than unity, the Jacobian of the Jacobian of f and <p with y^ is 7'educible. Let /=tta;^ <f> = b^\ yir^c^P {f<\>) = {ab)a^^-'h^''-\ Polarize once with respect to y, the result is ^■^' ^^^ = ^/i^^ ^"^^ ''"""' ^^"~' *^ -^ m + n--"2 ^"^^^ '*^"" ^^"" ^^- Hence (m + n-2)((/,<^),^/r) = {m - 1) (a6) {ac) a^""-- b:^"'' c/-^ + (?i - 1) (ab) (be) a^"'-' 6/-- c/"'. But 2 (a6) (ac) ^j^; c^ = (abf c^^ + {acf 6^' - (60)^ a^^^ and 2 (a6) (6c) a^ c^, = - {ahf c^ - {bcf a^' + {acf b^? *. Therefore = iaa;'""'6a,'*-''c/- = m — n w + n — 2 {aby c/ + (acY b^' - (be)- a^' m — n -^^^^^_^{f<f>r.ir+hAffr.<f>-H<t>,i^y'f-{^i^)- (ii) The product of two Jacobians may be expressed as a sum of products of covariants, ther^e being at least three covariants in each product; provided the forms of luhich the Jacobians are taken are all of order greater than unity. To prove this, we first establish a useful identity between symbolical forms. Consider the determinant (/,- Wl«2 ai b{^ bA b,' c,' C1C2 Cf til «9 it vanishes if -p = r . hence (ab) is a factor ; similarly [be), (ca) are factors. There can only be besides a numerical factor, which may * See Chapter I. § 22. 77] GORDAN S SERIES 79 be determined by considering the coefficient of a^hih^c^. The determinant is therefore equal to — {ah) (be) (ca). Hence 2 . (ab) (be) (ca) . (de) (ef) (fd) di - 2d,d. di" 62' Z6162 61 fi -2/,/, y,^ (adj (aef (aff (bdy (bef (b/y (XX). (cdf (cey (c/y \ In this identity let us put Ci == sCo, C2 = ^1 5 y 1 ^ '^■2> j2^ ^i- a^' a^a^ ai w bA bi c,' C1C2 ci Then 2 (ab) ax b^ . (de) dx e^ = (ady (aey Ox^ (bdy (bey bx' dx' e^ Consider now two Jacobians (f, (j)), (yjr, -)() ; where /=a^- = 6^", Then (/ 4>) ■ (^. %) = (a^) ax'""-' ^x'-' . (de) dx^-' e^'"^ = \ax'^-'bx''-'dx^-'ex'i-' (ady (aey a^ (bdy (bey bx' dx- ex- + i(<^.t)yX-i(0.%)V-f (XXI). For the sake of generality the forms /, <^, i/r, ^ have been supposed to be all different. The theorem is still true if two are equal to one another. In particular, it is true for the square of a Jacobian. 80 THE ALGEBRA OF INVARIANTS [CH. IV Another symbolical identity of interest is obtained thu-s : Form by means of the ordinary law of multiplication the product of the following vanishing determinants : ajttg ai Wh V CiC.2 «•/ d^d.^ di Se^ea fi -2/,/2 f^ 9% -'^9x9% 9i {agf (ahy (Jbgf {bhf {cgf {chf = (XXII). Hence if = 0. {aef {a/y {bey {hfy {cef {c/y {def {dff {dgY (dh) F=e^'', *=//', ■9=g/, X=A/ i (/, Ff (/, *)2 (/, •^f (/, x)2 ^ {<i>,Ff ((^,*)2 (0,<?)2 (0,x)2 (v^.i^)-'' (>/.,*)i' (>/.,*)2 {^,xy U,Ff (;^, *)2 (;f,*)2 (;t, X)^ Here, as before, the forms are not necessarily all diflFerent. 78. The expression for the product of two Jacobians may also be obtained as follows. We have (/ «^) X i-^, X) = («^) dx'^-'ho,''-' • {de) d,P-'e,9-' = (ab) (ae) a/^-^ft^^-^e^?-! . ^jr - {ah) {ad) a^'^-^b^^'-^d^P-' . x- and, by means of the identities of the type {ab) {ae) b^e^ = {ab^ e^ + (ae)'- bi - {hef ai, the right-hand side becomes \ (/ xr <f>f + ^ i<f>> n/x - i (/ ^)^ <t>x-i (4>, xT-ff as before. Thus if J be the Jacobian of y and - 2J^ = {fjy <f>^- + (</,, cf>yf^ - 2 (/, <f>)\f<i>. 79. Copied forms. If in the symbolical expression of a covariant 11 of a binary form F^^, the symbolical letters are taken to refer to another binary form <f>x^ where m^p, and 11 is multi- plied by aJ''-PbJ"-v ... 77-79] gordan's series 81 — where a,b, ... are the letters occurring in H — then the resulting form IT' is called a copied form. The original form 11 is called the model form. Ex. The Hessican of any binary form (l> = a^^^ = hx^^ is (a6)2a;,'»-26;,'»-2. It is formed on the model of the Hessian of the quadratic. The use which we are going to make of the idea of copied forms is for the case when the orders p and m of the fundamental quantics of the two systems are the same. The form ^ will be taken to be a covariant of degree two and order m of a binary form /x'*: then <^ = (/, ff<^ = {ahf" a^""--" ba,''--" = (f>x^-*^, 2n-4ia = m. On the model of the complete system for the general binary form F^^, we may construct a complete system for (f). If when this is done the symbolical letters <^i, <f)2, ... which refer to (f) are replaced by the symbolical letters a, h, ... which refer to yj each covariant of </> will become a covariant of/, which consists of a sum of sj'mbolical products instead of a single term. These separate products may be arranged in a series of adjacent terms, the difference between any two of which contains a factor of the foim (a6)^'^"'"'. Thus if we reject covariants of grade greater than 1(T, any one of the terms given by a covariant of ^ may be taken to represent this covariant. Before proceeding to a rigid proof of this statement, let us consider an example. The covariant of the sextic "z — "x — • • • is a quartic. Let us consider the Hessian of <^, It consists of a linear function of ten covariants of the following tj'pes : {abY{cdf{acfb^^d^\ {ahy{cdY{ac){ad)h:?c^d^, {aby (cd)* (ac) (bd) a^bxCxd^. Any one of these may be taken for the copied form, for each differs from the whole transvectant by transvectants of forms obtained from iab)*ax^bx% [cdfcx^d^^ by convolution. Hence when forms of grade higher than 4 are neglected, the whole transvectant and each of its terms are equivalent. G. & Y. 6 82 THE ALGEBRA OF INVARIANTS [CH. IV 80. Consider any symbolical product P, the symbols of which refer to in this product let us replace any particular letter <^<^' by a new variable y ; (we replace ^12 by y^ and ^u by — ^o). Let Py^n-w be the resulting expression, then = (Pj,2n-4<r , {aVf^ ay''-'"' by"-'"'y"-^ . Any term of this transvectant dififers from the whole trans- vectant by terms involving a factor (aft)"^"'"'"' : for since y is absent when the transvectant is expanded, the only kind of adjacent terms are of the form (aa) (y86) M, (ab) (/3a) M (see § 50). The above argument is not affected if we suppose that symbolical letters are present in P which do not refer to <f>. Hence we may replace each of the letters which refer to (f> in turn by letters which refer to /; and in doing this we may at each stage choose any one term to represent the whole expression P, provided that those terms which involve (aft)-""*"' are rejected. Thus taking the quartic invariant {abf(bcy(cay for model, we obtain the copied invariant of <\) = {abya^^b^\ viz. {(f>M^ i^ifk^iy ((t>3<f>iY- Any one of the sextic invariants (aby(cdy{efy(acy(dfr{eby, {aby(cdy{efy{ady{cey(fby, (aby (cdy (e/y (ac) (ad) (ce) (de) {/by differs from the invariant of (f>, by terms involving the factor {abf. Conversely in any co variant of a binary form aa;"= 6/'= ..., which has a symbolical factor (ab)-'', we may replace the letters a, b wherever they occur in the symbolical product by a single letter <f>, and remove the factor (aby'^ altogether: on the under- standing that forms of grade greater than 2a are being rejected, and that refers to the form (a6)-'^aa;"~-'^6a;"~-^ The reader will have no difficulty in verifying this statement. 80-82] gordan's series 83 81. G-eneralized Transvectants. Consider a product of any binary forms, /i(^) = ax'", My) = hy'\ fM = o^P. The result of operating on P with {m-\-^l) \ {n-\-v)\ (p-^^iiOUx Of^ O^ m\ ■ n\ 'pi ^^^y^^^z^^z,, where n.,= ^ - ^ *■' dx-idy^ dx^di/i ' after operation y and z being replaced by a;, is (aby {acY (chy aa;"*-^-'*6a;"-■^-''Ca;^-'*"^ This we define to be a generalized transvectant. Instead of taking forms each having a single symbolical letter, we may construct generalized transvectants of forms each of which has two or more symbolical letters. Thus we may replace a^;'" by The generalized transvectant may then be expanded as a linear function of certain symbolical products. Just as in the case of ordinary transvectants, any term of a generalized transvectant differs from the whole transvectant by lower transvectants of forms obtained from the original forms by convolution. We leave the verification of this statement to the reader. Further it is evident that any symbolical product may be regarded as a generalized transvectant. A copied form is then merely the same generalized transvectant with a new form taken for ground form. The theorem of § 80 — that any single term of a copied form, when a covariant of the second degree is the new ground form, may be taken to represent the whole form, provided that forms of higher grade than that of the fundamental ground form are to be neglected — is merely a particular case of that just enunciated. 82. Hyperdeterminants. When the forms of a generalized ti-ansvectant are all the same, it will be noticed that the trans- 6—2 84 THE ALGEBRA OF INVARIANTS [CH. IV vectant is entirely given by the differential operator. Thus a covariant of the binary form /= ax^, is completely defined by the operator ^^ xtj i^ xz ^''"zy Cayley used the notation to define such a covariant. These symbolical forms are called hyperdeterminants. Cayley introduced his calculus of hyper- determinants some years before the symbolical notation was invented by Aronhold. The hyperdeterminant notation was introduced for a single binary form merely for convenience. It is evident that it may be used perfectly well for covariants of two or more different forms. It is interesting to notice that the letters of a symbolical product may be regarded as differential operators. Thus if _ a p _ a_ _d "'-d^,' ^^"a.7i' '^^~afi' _ a „ 8 d dfa 07]^ 0?2 then {a^Y (ajY (7/3)" operating on the product of produces the covariant {ahY {acY (chY tta:"*"^"'* &«""*"" Cx^*""" multiplied by ml 111 p\ (m — \ — fi) \ {n — \ — v) I {p — fj, — v)\' provided that after operation ^, rj, ^ are each replaced by x. Further the operator (a^Y (<^yY (j^y olT-^-'^ /S^"-^-" 7xP-'^-'' acting on the same product produces the same covariant multi- plied by m\n\p\. In this case ^, 17, ^ all disappear after operation, so there is no question of replacing them by z. CHAPTER V. ELEMENTARY COMPLETE SYSTEMS. 83. Complete Systems of irreducible covariants. We shall devote this chapter to a detailed discussion of the invariants and covariants of single binary forms of the first four orders ; in particular, we shall obtain what are known as the complete systems of covariants for such forms. It has been observed, in fact, that the symbolical notation enables us to construct an infinite number of covariants of any form y*, but, as was first proved by Gordan, all these are rational integral functions of a finite number of covariants of f\ this finite number is said to constitute the complete system of irreducible concomitants, or, more briefly, the complete system of concomitants of the form. The general proof of Gordan's Theorem will be given in the next chapter. For the present we shall content ourselves with explaining easier methods of obtaining the complete systems in the simpler cases, and proving of course that such systems are actually complete. Inasmuch as every covariant can be expressed as an aggregate of symbolical products, we need only retain such as consist of one product in seeking for the complete system. 84. Linear Form. The discussion of a single linear form /=a* = &x = etc. presents no difficulty. For a symbolical product either contains a factor of the type (a6) or it does not. If it does so it is zero because (a6) = and if it does not it is simply a power of/. 86 THE ALGEBRA OF INVARIANTS [CH. V Hence every covariant of a linear form is a power of the form itself, or in other words, the form constitutes the complete system. Go7\ The same argument applies to any number of linear forms, for every symbolical product is a rational integral function of invariants of the type (ab) and co variants of the type ag.. Hence the complete system for n linear forms consists of the n forms themselves and the ^n{n — l) non-vanishing invariants of the type (ab). This result has already been established in § 37 where it forms the lemma preliminary to the proof of the fundamental theorem. 85. Quadratic Form. Suppose the form is /= cix^ = bx^ = Cx^ = etc. Then if a symbolical product contain no factor of the type {ab) it is a power of / ; if on the contrary it contains such a factor {ah) it can be transformed so as to contain {ab)- (§ 60), which is an invariant. Thus every invariant and covariant except ^x, {ahy can be expressed in terms of co variants of lower degree, hence by continued reduction we infer that every such form is a rational integral function of /(/./)- A, the latter being the discriminant of the quadratic. In other words, the form and its discriminant constitute the complete system. Ex. Prove that {ah) (ac) {bd) (ce) dxOx = jAy. (By interchanging a and h put the factor {abY in evidence.) 86. Before proceeding to the discussion of the cubic and the quartic we shall explain some general principles relating to the formation of the irreducible co variants of any given degree of a binary form. Suppose that the form in question is /= tta," = 6^" = c^'' = etc. then the only irreducible covariant of degree one is/ ■ 84-cSfil 84-86] ELEMENTARY COMPLETE SYSTEMS 87 Next, the only covariants of degree two are those of the type (a6)'-a^'»-'-6^''^-''; r = 0, 1, 2, ... 7i. If r be odd this covariant vanishes, and if r be zero it is reducible since it is equal to f^; the remaining forms corre- sponding to even values of r constitute the complete set of irreducible covariants of the second degree. Now assuming a knowledge of all the irreducible covariants of degree less than m we shall shew how to find the irreducible covariants of degree m. Suppose that the given irreducible forms are then any covariant of degree less than m is a rational integral function of / and the ^'s. Now a covariant of degree m is an aggregate of symbolical products containing m letters ; let C^ be one of the products and k one of the symbols involved, then Cm is a term in a transvectant where G-„i-\ is a product containing only ni — 1 letters, that is, it is a covariant of degree m — 1. Thus 6',, = (6',„_: , fy + S (C^, /)"', p'<p and C',„_i is derived fi-om Cni-\ by convolution. But C„,_j, C,rt_i being covariants of degree m— 1 are rational integral functions of the forms / 01, 0i, ••• 0r, i.e. they are aggregates of terms of the types f^m-i=/''0,'^' ... 0A of degree m — \. Consequently (7,„ is a sum of transvectants of the form (f^^n-i,/)", where of course p :|>- n. Since this is true for every separate term in a covariant of degree m it is true for the whole ; or, in other Avords, every covariant of degree m is expressible in terms of transvectants 88 THE ALGEBRA OF INVARIANTS [CH. V where Um-i is a product of the form and is of degree m — 1. Hence to find all the irreducible covariants of degree m we have to write down all transvectants of the form and reduce as many of them as possible. The remaining ones are the irreducible forms of degree m, for any covariant of degree m can be expressed in terms of them and covariants of lower degree. For example, in the case of a binary quintic the irreducible forms of degrees one and two are f,H= (aby a^ b^, i = {aby a^ b^. The only products of powers of these of degree two are f"^, H and i, so that all the irreducible covariants of degree three are included in (/^/)^ {Hjy, {ijy, where p ::|» 5 for the first two transvectants and p :^ 2 for the third, since i is a quadratic. 87. Let us now return to the transvectants {'Urn-.fy which we have to reduce as far as possible. That many of them are reducible follows from the following principles. Suppose that ^m-i = yW where Fand W are likewise products of powers of/, <^i, i/),, ... <^,., but of smaller degree than Um-\, and further suppose the order of W is not less than p. Then if T^ be any term belonging to V and Tn_ any term of the transvectant {W,fy, which is a possible transvectant because the order of W is at least equal to p, T^T., will be a term in the transvectant (Um-^jy. Hence {U,n-^,fy=T,T,+ t{[I,r^„fy■,p'<p and Ti, T^ being both covariants of degree less than m are expressible in terms of f and the ^'s. 86-88] ELEMENTARY COMPLETE SYSTEMS 89 Now in discussing the reducibility of the transvectants let us consider them in the order of their indices, e.g. we examine all those of index one before we proceed to any of index two, and so on. Then since Um-\ is a covariant of degree m — 1 it is an aggregate of products of the type Um-\ , and since p < p it follows from the equation that the transvectant on the left is completely expressible in terms of co variants of less degree and transvectants previously considered, or to put the matter briefly, it is reducible for it certainly cannot give rise to a new irreducible form. Hence the irreducible covariants of degree m can only arise from such transvectants {Um-^,fy for which Um-i has not a factor of order greater than p. Thus in the case of the quintic no transvectant of the type (A/y is irreducible because p 1f> 5 and the term f^ contains a factor / whose order is 5. In the general case if Um-i possess a factor W whose order is not less than n, then the product Um-i can give rise to irreducible covariants for no value of p ; it may therefore be neglected entirely in the search for irreducible covariants. As an application of this remark we note that if the order of one of the </)'s, say (f>p, be at least equal to n, then we need not consider the product of this form with any others. Finally if 0g be an invariant we may leave it out of account in forming the transvectants, because it would occur as a factor in each transvectant in which it appeared and so the transvectant would be reducible. 88. Irreducible system for the binary cubic. After these preliminary explanations the deduction of the complete system of a cubic presents little difficulty. 90 THE ALGEBRA OF INVARIANTS [CH. V Let the form be f=a^^^ hi = d = etc. then the only irreducible form of degree two is H={ahya^K = hi = h'i. To find the irreducible forms of degree three we note that the product /- is negligeable, so the only possible irreducible forms are {H,f), {Hjy. Now (H,f) = {ahy{ac)hci = -t, where t is an irreducible covariant, and {H, fy = {{aby a^ h, c^]' = (aby (ac) (be) c^ = — (be) (ca) (ab) {(ab) Cx} = — ^ (be) (ca) (ab) {(ah) c^ + (be) a^ + (ab) Cx], as we see by interchanging a, c and b, c and adding the results. Hence (H,/)'- = and the only new irreducible form is t. The prodvicts of degree three are /'. a/, t, of which the first two may be neglected ; hence the irreducible forms of degree four are included in (tj), (t,f)\ (tjy. Of these (t,f) is the Jacobian of a Jacobian and hence can be expressed in terms of forms of lower degree, § 77. In fact, to give the actual expression, we have {(H,f), ^] = i (/, ^fH-\ (H, iryf, whatever "^/r may be, since (H, fy = 0. Therefore - (tj) = i (fjy H-\ (Hjyf, or (t,f) = -\H\ Next (t,fy=^-\(ha)hxai,bx'Y = - (ha) (hb) (ab) axbx + X [(hay ax, 6*^}, since (ha) (hb) (ab) a^ bx is one term in the transvectant. 88-89] ELEMENTARY COMPLETE SYSTEMS 91 Now this term vanishes, and since we have Finally (t, ff = - {{abf (ac) h o^\ d^f = - {ahj (ac) (bd) {cdf = - A, so that (/, tf = A an invariant which proves to be irreducible. The products formed from/I H, t which are of degree four are /, f^H, ft, H\ and all except H- may be rejected because t and / are both of order three. Further {H^,fy is reducible unless p > 2 because H- contains the factor H whose order is two. Hence the only possible irreducible form of degree five is {H'\ff. But {H\fy = {K'h'^\ a^J = {haf{h'a)h'^ = — [{hd)- ax, h'a;-] = since (ha)'- a^ = 0, hence there are no irreducible forms of degree five, and in fact it is easy to see that there are no more irreducible forms. For if there were, the one coming next in ascending degree would be of the form (f^HHyjy. The only products that can lead to irreducible forms are f, t, H and H-, because when /3 > 2, H^ involves the factor H- whose order is greater than three ; but the transvectants arising from each of these products have already been considered, hence there are no more irreducible forms ; in other words, every invariant or covariant of the cubic is a rational integral function of / H, t and A. 89. Irreducible system for the quartic. If the form be J = Qjx =^ Ox = Ca- , then the irreducible forms of degree two are H = (abyax^hx- and i^{ab)*, H being of order four and i an invariant. 92 THE ALGEBRA OF INVARIANTS [CH. V The only product of degree two that we need consider is H, and hence the irreducible forms of degree three are included in {Hjf, (Hjy, (Hjy, {Hjy. Now (S>fy = W (ac) a^bJ'Ca,^ = - t where t is an irreducible covariant. {H,fy={{abyaJ^K\c,^Y = j\ (aby (acy bjc^' + j\ (aby (ac) (be) a^b^c^^ + ^ (aby (bey a^W = i (aby (acy ba^'c^' + f (aby (ac) (be) a^b^c^^ = i (aby (acy b^^c^^ + \ (aby c^^ [(acy b^' + (bey a/ - (aby c^^] = (aby(acyb^'cJ'-Uahy.Ca?, since the symbols are all equivalent. Then since, § 22, (aby c^' + (bey a^' + (cay b^* = 2 (aby (oAiy ba?e^' + 2 (bay (bey a^^d + 2 (cay (cby a^b^, we have (aby (acy b^c^' = \ (abf . e/. Therefore (H,fy = W- W= ¥f and is reducible. Next (Hjy = {(aby aj'b^', e^'Y = («^)' i^c)' W b^ for the two terms in the traiisvectant are equivalent. By inter- changing a and c it follows that (H,fy = 0. Finally (H,fy = {(aby a A', Ca^'Y = io,by (bey (cay =j an invariant ; hence the only irreducible forms of degree three are t and j. The only product /"H^ty of degree three that we need consider is t, and the only possible irreducible forms of degree four are therefore (tjy, (tjy, (tjy, (tjy. As a matter of fact these are all reducible. 89] ELEMENTARY COMPLETE SYSTEMS 93 To calculate them we use the series of ^ 54, with the scheme f H f 4 4 4 r 1 which gives or '4 — r\ /I /4 - rW =2^v^^i(/,/)-^)-'. If r = 1 we find since (f, Hy = ^if, hence (t,f) = ^H"-j\if'. Ifr=2, (t,fy + {{f,Hyj] + j'A(fHfJY and since (f,HY = ^if, {fHf = 0, this gives (^./)' = 0- From r = 3, (t, fy + f {(/> sy-jY + A {(/ ^)^/} + i (/ ^)^ •/ = Ufy,H} + ^(/,/y.H, or a/>'^-i^•5^ + lJ/=i^i^, on putting in the values for the transvectants off and H. Thus {t,fy=i{iH-jf). The series does not apply when r = 4 because then r + 1 > 4, but it is easy to calculate (tjy directly or as in § 94. 94 THE ALGEBRA OF INVARIANTS [CH. V For {(ah) a^'h^\ K*Y = («^0 («&)' ^* V + ^ \{ahy a^%^\ b^'Y + fji {(ahy a^K, h^*Y + ^ {i<^hy, b^'}, while (aby {ah) b^^h^^ = [{aby a^b^, V} = 0, {ahya^'h^' = iif; (/,/>' = 0, {ahy a^hx = 0, hence (^,/)* = 0. Hence there are no irreducible forms of degree four. In fact there are no more irreducible forms, because the next in order of degree would be of the form (/"HHy,/)". Now since /, H, t are each of order four at least, irreducible forms can only arise from products containing each of the three by itself, and all these, viz. ifjy, (sjy, (t,fy have been already considered. Hence every invariant or covariant of the quartic is a rational integral function of/, H, t, i and j. 90. Quintic. To illustrate still further the method of this chapter we shall apply it to some extent to the binary quintic. The covariants of degree two are (/,//= ZT and {f,fy = i. The products of powers of /, H and i which are of degree two are /^ H and i, and to find the covariants of degree three we have to consider transvectants of these three forms with f. The transvectants arising from /-' may be neglected, and hence we are left with {H,f)\ (H,/)\ (Hjy, {H,f)\ (Hjy, {i,f)\ (i/y as the only possible irreducible covariants of degree three. Now of these (H,fy = {{aby a^b^\ c^'^Y 89-90] ELEMENTARY COMPLETE SYSTEMS 95 and involves a term ^ {aby (ac)' a^bai'c^', .-. (Hjy = (aby (acy a^b^^c^^ + \ {(aby a^b^, c^»|« = ^axKcx {{ciby (acy b^Cx' + (6af (6c)- c^^ai + {cay (cby ax%^] + \if = ^a^hcx I {(aby Cx* + (bey a^' + (cay 6^^} + \if = (\ + ^)if, so that (H,fy is reducible. Again (H, fy = \(aby a^b^, c^ and contains the term (aby(acybxHx\ This terra can be transformed so as to contain (ocV and hence must be a multiple of (aby (ac) b^c^* since the letters are equivalent. .-. (H,fy = X (aby (ac) b^Cx* + fi {(aby a^bx, c^»| = (\ + fM)(i,f). Further (H, fy = (aby (bey (acy a^b^c^ + \ a fy and (H,fy = (aby(bcy(acya, = i (bey (cay (aby {(bo) a^ + (ca) b^ + (ab) c^] = 0. Finally (i,fy is an irreducible form and (i,fy = (aby(ac)(bc)c,^ = - i (be) (ca) (ab) {(aby c^' + (bey a^' + (cay b^'} = - (bey (cay (aby a^b^c^ (§ 22). Hence the only irreducible covariants of degree three are (H,f) = t,(f,i) and (/, iy = - (bey (cay (aby a^b^c^ = -j. The reader may now hnd the irreducible forms of degree four and verify the result by reference to the chapter on the quintic. It will be seen at once that the method leads to much labour, that the reduction processes are not easy to discover, ai^d, when we mention that for the quintic we have to proceed step by step 96 THE ALGEBRA OF INVARIANTS [CH. V until we get to degree 18 before the irreducible system is obtained, the impracticability of these methods in dealing with forms of order greater than four will be at once admitted. 91. Further Theory of the Cubic. Syzygy among the irreducible forms. There is an identical relation con- necting the irreducible concomitants of the binary cubic — the simplest example of what is known as a syzygy among the covariants of a single binary form. In fact since t is the Jacobian of/ and H we have, § 78, - 2t^ = (f,ffH^ + {H, Hyp-^iHJfHf. Now {H,ff=^{ahfa^ = (H, Hy = [{ahf a^h, {cdf cM' = W (c^)' («c) (bd) = A, for although there are four terms in the transvectant they are identical in value, and we have the relation required. 92. Since every covariant of the cubic is a rational integi-al function of/, H, t and A it follows that all expressions derived by convolution from products of powers of/ JET and t can be expressed in terms of/ H, t and A. The form of the expression can be easily inferred by con- sideration of its degree and order, but the actual determination of the coefficients may be a troublesome process. As an example consider the Hessian of t, i.e. {t, tf. It is a covariant of degree six and order two since t is of degree three, and the only product of / H, t, A fulfilling these conditions is ^A. We at once see that (t, tf is a numerical multiple of ZTA. The reader may calculate the actual value directly by using the series of § 54 — we give an alternative process. Let J=(t,f), then - 2J^ = {t, typ + if, ff P - 2 (/ tyft 90-93] ELEMENTARY COMPLETE SYSTEMS 97 But {t,f) = ^H\ {f,fy=H, (f,ty=o and t' = -\H^-l Ap ; therefore -^H' = (t tfp ^H{-\E'-\ A/^) ; that is {t,tfp=\ApH or {t,ty=lAH. Again, consider the symbolical product {ahy (ac) (bd) (cd) Cxdx which represents a covariant of degree four and order two. Since there is no product of this degree and order the covariant in question must vanish identically. To verify this we remark that, on interchanging a, b and c, d, the expression (aby (ac) (bd) (cd) c^d^ changes sign ; hence it vanishes. Ex. (i). Calculate the following transvectants in terms of/, II, t, A, viz. {ff, H), {H, H)% it, t), {t, t)\ {t, tf, (IT, t), {H, tf. (The only one presenting any difficulty is {H, t) and this is the Jacobian of a Jacobian ; its value is — ^ A/.) Ex. (ii). Shew that any symbolical product involving the factor {aKf vanishes identically. Ex. (iii). Shew that {m,pf = 0, {H^fif = 0, {H\P)=^H'^f{HJ)=-H^ft. 93. Further Theory of the Quartic. As in the case of the cubic, the square of the covariant t can be expressed rationally in terms of the remaining forms. In fact we have, § 78, - 2t^ = (f,ffH^-2(H,fyHf+(H, Hfp, while {f,fr = H, (HJf = iif, so that it only remains to calculate (H, H)'\ G. & Y. 7 98 THE ALGEBRA OF INVARIANTS [CH. V Now using the series of | 54 with the scheme ( 4, 4, 4 J \ 0, 2, 2 / we have (') (') (') (') 'f7') 1'7'' or {H,Hr+[{f,f)\H]^^{fjy.H = {(/ ^)^/}^ + ((/, Hr,f\ + 1 (/ Hy.f, that is (£r, Hy + liH = ii (/ /)^ + ^ j/, since (/, H)'^ = etc. Hence {H, Hf = iJf-^iH. Consequently which is the syzygy required. 94. We shall illustrate the reduction of covariants of the quartic by calculating the values of the transvectants of /, H, t, taken two together. The transvectants of f with itself, H and t have already been found. As regards the trans vectant (H, Hf we remark that it vanishes when r is odd and (H,Hy = yf-iiH. There only remains (H, Hy. This is equal to {(ahy a^'h\ h^*]* = (ahy {ahy (bhy = {{ahy a^'h^\ hJ'Y = [{H,fy,fY=ii{f,fy = li\ 93-94] ELEMENTARY COMPLETE SYSTEMS 99 To calculate the transvectants {t,Hr, r>>3, apply the series of § o4 with the scheme / H, f, H^ ( 4, 4, 4 ) \ 0, r, 1 J and we have ?)(i),„ ..„,„,. .^r I On taking r=l, 2, 3 successively and putting in the values of the transvectants (H, HY'^^ we find (t,H)=if{iH-jf) {t, Hy = For {t, Hy, the scheme / H, f, H^ 4, 4, 4 V 1, 3, 1 / must be used ; or else as in the case of (t,fy it is easy to see that {t,Hy = o. To find (t, ty we apply the series with the scheme { 4, 4, 6 ) \ 0, 2, 1 / which gives {(/ H), tY + {{/, Hy, t] + j\ if, Hy t = u ty, h] + § (/ ty h or {t,ty + \{f,t) = l{f,ty.H. On substituting the values for {t,f) and {t,fy we find (t,ty = ujfff-^^H^'-^iT)' 7—2 100 THE ALGEBRA OF INVARIANTS [CH. V For {t, ty we use the scheme //' ^' «\ I 4, 4. 6 ) \ 1, 3, 1 / leading to {t,ty+l(f,ty={{t,fy,BY -\is,Hy-iif,Hy and hence {t, ty = 0. Finally (t, ty = {{t, fy, HY = i i (H, Hy - ij (/ Hy Ex. (i). Deduce the value of {t, tY from the relation -2{(«,/)}2=(«, typ+(f,fyt'-2(f, tfft. Ex. (ii). Apply Gordau's series to calculate {t, t)^ for the cubic. Ex. (iii). Prove that Ex. (iv). Prove that for the quartic {{ff,iif,H}=+yt {{H,H)\HY = \f-^z\ Ex. (v). Prove that the Hessian of the Hessian of the Hessian of a quartic / is Ex. (vi). Calculate the values of {{t, t)'^, ty for r=l, 2, 3, 4, 5, 6. Ex. (vii). If a quartic / be the product of a cubic by one of the linear factors of its Hessian, then (/,/)*=o. CHAPTEE VI. GORDAN'S THEOREM. 95. We have already referred to Gordan's theorem which asserts the existence of a finite complete system of covariants for any binary form, and, in fact, we have illustrated the truth of the theorem in obtaining the complete systems for the quadratic, cubic, and quartic. Our previous method is of little practical utility in dealing with forms of order greater than four, but a comparison between it and the procedure of Gordan may not be without value as a primary indication of the salient features of the latter. In the last chapter covariants were classified according to their degree and we shewed how to obtain those of degree m by transvection from those of less degree. In Gordan's investigation covariants are classified according to their grade — the grade being a definite even number associated with any symbolical product, § 61, and all covariants of grade 2r are obtained by transvection from those of inferior grade together with some of grade 2r. The advantage of using the grade is that no covariant can be of grade greater than n ; accordingly, the number of steps in the process is small, whereas there being no limit, cb priori, to the degree of an irreducible covariant, and the actual degree reached by irreducible forms of quintics, etc., being very high, the number of steps in the other process is uncertain and at the best large. As will be seen later, on the other hand, the transition from grade 2r — 2 to grade 2r is commonly much more difiicult than that from one degree to the next higher. Several preliminary propositions are necessary before we can undertake the actual proof of the existence of the complete system ; these we now proceed to explain. 102 THE ALGEBRA OF INVARIANTS [CH. VI 96. The first lemma required belongs to that branch of the theory of numbers known as Diophantine Equations. For the sake of clearness we shall begin by giving an illustration. Consider the homogeneous linear equation 2x + 5y = 2z ; it is easy to see that the number of solutions in positive integers is infinite. Moreover, if and so=p\ y = q, z = r\ be two solutions, then x = 'p-\-'p, y = q + q', z = r + r is also a solution. We shall call this latter the sum of the former two solutions, and when any solution can be written as the sum of two smaller solutions (throughout we deal only with solutions in positive integers), it is said to be reducible. Otherwise a solution is irreducible, and the important fact for us is that the number of irreducible solutions is always finite. Thus for the equation above the only irreducible solutions are x = S, y = 0, z = 2; a; = 0, y = S, z = 5; a;=l, y = 2, 2^ = 4; x=2, y = l, z = S. In fact if a; > 3, then z>2, and the solution can be reduced by means of a; = 3, y = 0, ^ = 2 ; whereas if y > 3, then z>5, and the solution can be reduced by means of « = 0, 2/ = 3, ^= 5 ; thus in an irreducible solution neither x nor y can exceed 3, and, as the number of remaining possibilities is finite, the irreducible solutions can be easily found by trial. By continually reducing a given solution, say a; =p, y = q,z = r, we can express it in terms of the irreducible solutions, that is in the form p = SX + v + 2p ) q = Sfi + 2v + p i (A), r=2\ + 5fi + 4iv + Sp) where \, fj,, v, p are positive integers : r 96-97] gordan's theorem 103 e.g. take the solution x = oO, y = 7, z = 45, reducing by means of a; =3, 2/ = 0, z = % a;=16.3 + 2, 2/ = 7, ^=16.2 + 13; then reducing x' = 2, y' = 7, 0' = 13 by means of a; = 0, 2/ = 3, 2^ = 5, a;' = 2, y = 2 . 3 + 1, / = 2 . 5 + 3, and the remaining part a^' = % 2/"=l. /' = 3 is irreducible. Hence in this case we have \=16, fi=^1, v = 0, p = l. Of course if we substitute the expressions in (A) for x, y, z the equation is satisfied identically ; the important point is that evei^y positive integral solution can be written in the form there indicated. 97. The idea of reducibility can be extended at once to any number of linear homogeneous equations, for the sum of two solutions is always a solution, and we may enunciate our first lemma as follows : The number of irreducible solutions in positive integers of a system of homogeneous linear equations is finite. Consider first a single equation, a^Xi + a.x^ + . . . + amXm = &i 3/1 + ^2^2 + . • • + hnyn, connecting the x'b and y^, where the coefficients a, h are positive integers. If the two solutions ^i^^Si) '^2^62' ••• '^m~ ^m> yi^^Vi' y-2^V2t ••• yn^^Vn'} ^1 = ^1, ^2 = ^2, . • • ^m = ^m, 2/1 = Vi, 1/2 = V2, • • . yn = Vn, be typified by 104 THE ALGEBRA OF INVARIANTS [CH. VI respectively, then also typifies a solution and this latter is reducible. First the equation has mn solutions of the type with the rest of the variables zero. Next suppose that in a solution one of the xs (say x-^ is greater than 61 + 62+ •■• +&«, then the right-hand side of the equation must be greater than ai(&i + &2+-.. + 6n), i.e. 61 (2/1 - Oi) + 62 (2/2 - ai) + . • . + hn (y« - aO > 0, so that at least one y must be greater than a^. Let yr>(h, then the solution in question is reducible by means of the solution with the other variables zero. Hence (61 + 62 + . . • + 6„) is an upper limit to the value of any X in an irreducible solution,. similarly (oi + ^2 + • • • + ^m) is an upper limit to the value of any y ; but the number of solutions reducible or irreducible subject to these restrictions is manifestly finite, therefore d fortiori the number of irreducible solutions is finite. If the irreducible solutions be typified by «? = «!, 2/ = /3i ; sc = (h, y = ^2] « = «P' y = ^i» then by continued reduction any solution can be expressed in the typical form a; = ^iffi + ^2«2 + . • • + t^ap y = t,^, + t^^,+ ...+t,/3„ where the t's are all positive integers. Suppose now we have a second equation of the same nature between the variables ; then on replacing the x's and y's by their 97] gordan's theorem 105 values in terms of the ^'s the first equation will be satisfied identically and the second equation will become a linear equation between the ^'s with integral coefficients. Hence by the above reasoning every solution of the equation among the ^'s may be written in the typical form ^=T,7i+ 2^272 + ... +r,7<., where t = yi, ^ = 72, ••• t = ya^ typify the irreducible solutions, and the T's are all positive integers. Now substitute these values for the t's in the expressions for the xs and y's and we find at once that where the /c's and X,'s are fixed positive integers. Thus the only possible irreducible solutions of the two equations are those typified by X = K^, y ^= \,i\ X =^ K2, y ^^ A<2 J ... X ^ K(r , y ^ A,(, , for every other solution can be expressed as a linear combination of these. If we had a third equation, on substituting for the xs and y's their values in terms of the T's the first two equations would be satisfied identically, and the third would become a linear equation among the T's. Then this equation in turn has only a finite number of irreducible solutions, and hence, reasoning exactly as before, we should find that the three equations given have only a finite number of irreducible solutions. The process can be manifestly extended to any number of equations, and hence our theorem is established. A formal proof by induction from (r — 1) equations to r equations could of course be easily given. Ex. To find the irreducible solutions of the two equations x-\-w=y + z)' The irreducible solutions of the second equation are easily found since no letter can exceed 2 ; they are x=\, y = l; x = \, 2 = 1; y = l, w=\; z=l, w = l; variables not naentioned in a solution being zero. 106 THE ALGEBRA OF INVARIANTS [CH. VI Hence the general solution of the second equation is a;=a + b, i/ = a + c, z=b+d, w = c+a, where a, 6, c, d are positive integers. The first equation now becomes and for an irreducible solution a, 6, c cannot exceed 2. On trial we find the following irreducible solution a = 2, c?=5; 6 = 2, d=\; c = l, d=\; a = l, 6 = 1, d=Z. Hence the general solution is a=2a + S, 6 = 2/3 + S, C=y, c?=5a+/3 + y + 38. These values for a, 6, c, d give a7 = 2a + 2i3 +2S\ y = 2a + y + « = 5a + 3j3+ y + 4S 20 = 50+ /3 + 2y + 3S, as the general solution of the two equations. The only possible irreducible solutions are accordingly ^=2, y = % z=5, w=5 x=2, y=0, 2 = 3, w=l x=0, y=l, 2=1, w = 2\ x=2, y = \, 2=4, xo=Z} Of these the first is the sum of the third and fourth, while the foui-th is the sum of the second and third, so the only irreducible solutions are the second and third. In other words any solution of the two equations may be written in the form x=2p, y = q, z = 3p + q, w=p + 2q, where p, q are positive integers. Ex. (i). Prove that the equation lx + 4y—3z has four irreducible solutions and that every solution of the two equations 7x + 4y = 3z, z+5w=2y can be written in the form x=2a + c, y = 7a+l5b + Uc, 2 = 14a + 206 + 17c, w = 2b + c. Ex. (ii). Find the number of irreducible solutions of the equation 11 •" 2 2 ■" • • • "t" n n ^~ ^3C^ the a's being positive integers. Ans. If all the a's are even there are n irreducible solutions, if r of rir—V) the a's are odd there are « + — solutions. In an irreducible solution at the most only two of the letters on the left are different from zero. 97-99] gordan's theorem 107 Ex. (iii). Prove directly that in an irreducible solution of the two equations hyi+hy2+ ••• +Kyn=y+z X is less than the greatest a, and y is less than the greatest h. 98. System of forms derived by transvection from two given systems. Consider two systems of binary' forms in the same variables a^i, x.^, viz. J-i, A.2, ... Am, of orders a,, a^, ... a^ respectively and Bi, B^, ... B^, of orders h^, b^, ... bn respectively; we suppose each form written symbolically and denote by U, V two products of the types A,-^A,-^ . . . Am'^'n, B/^Bl^ . . . B,fn wherein all the exponents are either zero or positive integers. The system C is said to be derived from the systems A and B by transvection when it includes all terms in all transvectants of the form {U, V)y. It is clear that some of the members of the system C are reducible, that is they can be expressed as rational integral functions of simpler meijibers of that system — in fact, if then there are many terms in the transvectant {U, V)y which are products of two terms, one belonging to the transvectant and the other to the transvectant (U,, V,)y^. 99. We can now enunciate and prove our first theorem, viz. The number of transvectants of the form ( U, V)y which do not contain reducible terms is finite. * It is of course assumed that y^, y^ are such that these transvectants are possible, e.g. y^ must not exceed the order of V-^. 108 THE ALGEBRA OF INVARIANTS [CH. VI For suppose that any term of the transvectant contains p symbols of the A's, not in combination with a symbol of the B's and a symbols of the B'& not in combination with a symbol of the J.'s, then we have because each side of the first equation, for example, represents the total number of the symbols of the forms A which occur in the product U. Now to each positive integral solution of the above equations in a, y8, p, o", 7 there correspond definite products U, V and a definite value of 7 and hence a unique transvectant. But as we have already remarked if the solution corresponding to (17, V)y be the sum of those corresponding to (U^, Fi)^' and {17.2, ^■2)'^^ then (U, V)y certainly contains reducible terms. Hence trans- vectants corresponding to reducible solutions always contain reducible terms and inasmuch as the number of irreducible solutions has been proved to be finite it follows that the number of transvectants not containing reducible terms is finite. 100. In actually finding the transvectants which do not contain reducible terms we may use the equations (I), but it is generally easier to proceed directly. Suppose that the system A contains the single form /= a^^ and the system B the single form i = hi, then we have to consider transvectants If 7 > 2/3 this transvectant vanishes, if 7 < 2y3 — 1 it contains terms of i (/", i^~^)t ; hence for an irreducible transvectant we must have 7 = 2/3—1 or 2/3, In the same way 7 :^ 5a and -^ oa — 4. Again if a > 2, then for an irreducible transvectant 7 > 10 and hence /3 > 4, so that some terms may be reduced by means of (/^ i^Y. Thus we need only consider a = 0, a = 1 and a = 2. For a = we have i. For a=l we have/ (/i), {f,i)\ {f,i-)\ {f,ij, {f,i^)\ 99-101] gordan's theorem 109 For a = 2 we have {f\ i'Y, (/^ iy, {f\ i*y, (/^ i% Now {f^, iy contains terms which are products of a term of (/, i'^ and (/, i)'^ and a like argument applies to (/^ i% (A i% (A i'f> so the only transvectants not containing reducible terms are / i if, i), if, if, (/ i% (/, i')\ (f, iJ. (A> iT- Ex. (i). If/ be any form of order 2n + l, then the transvectants which do not contain reducible terms are 2?i + 4 in number. Ex. (ii). Find the corresponding result when / is a form of even order. Ex. (iii). The only transvectants where /[=aa;* and/2=6a;^, which do not contain reducible terms are /i, /2, h (/i, i\ ifi, ^7, (/i> i'r> (/i' ^')s (A, i), (/2> if, (A, i')\ if2% i'f- 101, Definition. The system of forms A is said to be complete when any expression derived by convolution from a product U of powers of the forms A is itself a rational integral function of the A's. Thus for example the system of forms H = (abfa^bx t — {ahy {ca) hxC^ A = {aby(cdy{ac){bd) is complete because any expression derived in the above manner is a covariant of/ and therefore a rational integral function of /, H, t and A. Again the system H and A included in the above is itself complete. More generally the system A is said to be relatively complete for the modulus G consisting of a number of symbolical deter- minants when any expression derived by convolution from a product i7 is a rational integral function of the ^'s together with terms involving the modulus G. 110 THE ALGEBRA OF INVARIANTS [CH. VI Thus the system consisting of a single form is relatively complete for the modulus (aby, since any expression derived by convolution from a power of f can be transformed so that a factor (aby occurs in it. Again for a quartic the system /= a^^ H = (aby aA^ t = (aby (ca) aj)^^c^\ j = {bey {cay {aby is relatively complete for the modulus {aby, for all covariants of / are rational integral functions of f,H, t,j,i, where i = {aby. We may extend our definition of relative completeness still further : a system A is said to be complete relatively for several moduli Gi, 0^ ... when any expression derived by convolution from a product C is a rational integral function of the A's together with terms involving one at least of the moduli G^, G^ It will be seen later (or it can be verified without difficulty) that in connection with any quantic a^^ = bx^ ... the single form H = {aby a^^'-^b^''-'' is relatively complete with respect to the modulus {aby except when n = 4. If n = 4 the complete system worked out for the form H shews that any expression derived by convolution firom a power of H is a rational integral function of H together with terms involving i or j. That is H is relatively complete for the two moduli {aby and {bey {cay {aby. It will be noticed that a complete system is relatively complete for any modulus or systems of moduli. 102. The system G derived by transvection from two given systems contains an infinite number of forms, but it is said to be a finite system when all its members can be expressed as rational integral functions of a certain finite number of them. More generally it is said to be relatively finite for a given modulus G 101-103] gordan's theorem 111 when every member of C can be expressed as a rational integi"al function of a certain finite number of them together with terms all of which involve the modulus G. For example the number of covariants of a binary cubic is infinite but inasmuch as every one is a rational integral function of/, H, t and A the system of forms is said to be finite. Again it will be seen later that every covariant of the binary w-ic can be expressed in terms of/, H, t, where t = {ahf (ca) a^-^h^'-^c^-"- together with terms involving the factor {ahf. We should state this fact thus — The system of covariants of a binary n-\Q is relatively finite for the modulus {ahf. 103. Theorem. If the systems of forms A and B are both finite and complete, then the system derived from them by transvection is finite and complete. (a) The system is finite. In the proof of this theorem we shall consider the transvectants {U, V)y in a certain order defined as follows : — (i) Transvectants are taken in order of ascending total degree of the product UV in the coefficients of the forms involved in A and B. (ii) Those for which the total degree is the same are taken in ascending order of indices. Further than this the order is immaterial. With this convention let T and T' be any two terms of the transvectant {U, V)y, then {T-T') = L{U,T)y' where 7' < 7 and U, V are derived by convolution from U, V respectively. 112 THE ALGEBRA OF INVARIANTS [CH. VI But since the systems A and B are complete U = F{A\ where F{A) is a rational integral function of the J.'s, that is, an aggregate of products of the type TJ, and ^ (jB) a similar function of the 5's. Thus {iJ^yy can be expressed as the sum of a number of transvectants in each of which the index is less than 7. By hypothesis all such transvectants have been examined before the one now under consideration and hence if all the (7's derived from previously considered transvectants can be expressed in terms of Cj, Cj, ... Cy, then all C's up to and including those derived from can be expressed in terms of Ci, C^a, ... Cry T, where T is any term of the last transvectant. But if the transvectant contain a reducible term, say T=T^T^, then inasmuch as T^, T^ must both arise from transvectants previously considered no term T need be added to the system Cj, C2, ... Gf. Thus in gradually building up a system of C's in terms of which all O's can be expressed we need only add a new member when we come to a transvectant containing no reducible term and then we need add only one new member. But the number of transvectants containing no reducible term is finite and hence a finite number of (7's can be chosen such that every other is a rational integral function of these, that is the system G is finite. Remark. A set of C's in terms of which all others can be expressed rationally and integrally can be chosen in various ways, for any term may be selected from each transvectant containing no reducible terms. Further since the difference of two terms of a 103] gordan's theorem 113 transvectant can be expressed by means of terms of transvectants previously considered we may, instead of choosing a single term from any transvectant, take an aggregate of any number of such terms or even the transvectant itself, and it will still be true that every member of G can be expressed as a rational integral function of the members of our finite system. (6) The finite system so constructed is complete. i-iet Oj, L/2, ••' 0,. be the finite system, then we have to prove that an expression W derived by convolution from any product of the form is a rational integral function of C^, C^, ... C^. Suppose that W contains p determinantal factors in which a symbol belonging to a form A occurs in combination with a symbol belonging to a form B. Then W is a term in a transvectant (u, vy, where U contains only symbols of the A's and V only symbols of the B's, so that U is derived by convolution from a product U of the ^'s and V is derived by convolution from a product V of the B's. Thus W={u, vy + tiuvy, where p < p and UV are derived from U, V by convolution and therefore ultimately from U, V. Now U=F(A), V=^{B), accordingly W can be expressed as an aggregate of transvectants of the form (U, vy. But we have just proved that every term of such a transvectant is a rational integral function of the C's and consequently W is also a rational integral function of them. Hence the system is not only finite but complete. G, & Y. 8 114 THE ALGEBRA OF INVARIANTS [CH. VI 104. Theorem. If a finite system of forms A, all the members of which are covariants of a binary form f include f and be relatively complete for the modulus H ; if further, a finite system B be relatively complete for the modulus G, and include one form Bi whose only determinantal factors are H, then the system C derived by transvection from A and B is relatively finite and complete for the modulus G. As an example of the theorem let A consist of f=cix= bx\ and B of the two forms E = {aby aj)^, A = {abf (ac) {bd) {cd)\ Then A is relatively complete for the modulus (a6)^ § 88, and B is absolutely complete, being the complete system of the Hessian of the cubic ; hence according to the theorem the system derived by transvection should be absolutely complete. This is obviously true, for the new system contains /, H, t. A, where t = {fH) = -{ohy{ac)b^c^\ and every possible member of the derived system is a covariant of /, therefore they are all rational integral functions of f H, t. A, which constitute the complete system of the cubic. 105. Lemma. If P be derived by convolution from a power off any term in the transvectant (P, vy can be expressed as an aggregate of transvectants of the type {U, vy in which the degree of U is at most equal to that of P. (Throughout we shall use U, V as typical symbols for products of powers of the forms of A and B respectively.) This statement is manifestly true when the degree of P is zero ; assuming it true when the degree of P is less than r we shall establish it when the degree is r. In fact if T be a term in (P, vy, T={P,vy+X{P, vy- 104-105] gordan's theorem 115 and since P, P are derived by convolution from a power of the form f which is contained in A, P=^F{A) + HW*, P = F'{A) + HW', while V=^{B) + GZ=^{B), mod a Hence T can be expressed as the sum of three parts ; (i) transvectants of the type [F{A), ^{B)Y the degree of F{A) being r ; (ii) transvectants of the type {Q, Vy, where Q is of the same degree as P and contains the factor H ; (iii) terms containing the factor 0. Now Q can be derived by convolution from where s is less than r the degree of P ; therefore any term in can be derived by convolution from and is expressible in the form S(P',5rr), where P' is derived by convolution from /* and is of degree less than P. But by hypothesis every terra in these transvectants can be expressed as an aggregate t{U, vy, modG^, for 57F=^(P), modG^, where the degree of U is at most equal to s and therefore less than r. On referring to the expression for T we see that T can be written in the form ^{U, vy, modG; consequently the statement in the lemma can be completely established by induction. * HW simply means a symbolical product containing the factor H. 8—2 116 THE ALGEBRA OF INVARIANTS [CH. VI Cor. If the product P contain the factor H, then any term in (P, Vy can he expressed in the form where the degree of U is less than that of P. For P is now of the fonn Q just discussed, and any term in a transvectant (Q, vy can be expressed as a sum t(U,vy in which the degree of U is at most equal to s which is less than the degree of P. 106. The proof of the theorem is now the same in principle as that in § 103. The transvectants are considered in the following order. (i) In order of ascending degree of UV in the coefficients of/ (ii) Those for which the degree of UV is the same are taken in order of ascending degree of U. (iii) Transvectants for which these two degrees are the same are taken in order of ascending index. Further than this the order is immaterial. If T and T' be two terms in {U, vy^ _ then T'-T=t{U, Vy\ where v <v. But U=F{A) + liW, therefore T'-T=t[F{A), ^{B)Y + ^[HW, <^{B)Y, modG. Transvectants of the type {FiA),^{B)y have been previously considered, for the degree of F(A) is the same as that of U and v <v; further by the lemma transvectants of the type [HW^^iB)]"' 105-106] gordan's theorem ' 117 can be expressed in the form where the degree of U' is less than that of HW, i.e. less than that of U. . Thus T'-T can be written 2 ( U", Vy + t(U', Vy, mod G, where the degree of U" is the same as that of U and v' < v, while the degree of U' is less than that of U. Hence if all terms of transvectants considered previously to {u, vy can be expressed rationally and integrally in terms of Gi, C2, ... Gp (except for terms involving G); then all terms of transvectants up to and including {U, vy can be expressed in the form F{C„G,,... Gp,n modG, where T is any term of the last transvectant. If the transvectant (u, vy contain a reducible term we may suppose it to be T, and since T—T^T^ where T^, T^ are terms of former transvectants, there is no need to add the term T to Cj, Cg, ... Cp. It follows that in constructing a system of (7's in terms of which all (7s can be expressed we have to add a new member only when we come to a transvectant containing no reducible terms and then one only. The number of transvectants con- taining no irreducible terms is finite, § 99 ; hence if G^,G^, ... Gg be a series of terms one from each of this finite number of transvectants, any other member of the system G derived by transvection from A and B can be expressed as a rational integral function of G^G^, ... Gq together with terms involving the factor G; in other words, the system G is relatively finite for the modulus G. Next the system (7i, G^, ... Gq is relatively complete with respect to the modulus G. 118 THE ALGEBRA OF INVARIANTS [CH. VI For any term T derived by convolution from may be regarded as a term in a transvectant (u, vy, where U is derived by convolution from a product of the A's and V from a product of the B's. Hence T can be expressed as an aggregate of transvectants {U,vy- while U=P can be derived by convolution from a power of / and V=^{B), modG; therefore T^^\P,<^ {B)Y, mod G, = X (P, vy, mod G, = t(U, vy, mod G. (Lemma.) Consequently, as has just been proved, T = F{G„G„... Gq), modG, and the system is complete. 107. Cor. I. If the system B is absolutely complete, then the system derived by transvection from A and B is absolutely complete. Cor. II. If the system B is complete for two moduli G and G' and contains a form whose only determinantal factors are H, then the derived system is complete for the two moduli G and G'. To prove this we have only to write B = F{B), modd {G,G') instead of B = F(B), mod (G) at every stage of the foregoing proof. 108. Gordan's Theorem. These long preliminary ex- planations are now at an end and the actual proof of the theorem does not present much difficulty. Every co variant of a binary form /=a^« = 6^" = etc. is either a power of / or else contains a factor (aby, and hence the form f itself is a complete system with respect to the modulus (aby. lOG-109] gordan's theorem 119 Assuming now that a system of co variants containing / and relatively complete for the modulus (ab)^ can be found we shall shew how to construct a system also containing f and relatively complete for the modulus (a6)^+^ The system relatively complete mod (aby'' is called -4^-1, and since every covariant can be derived from / by convolution it is a rational integral function of the forms in A^^i except for terms involving the factor {obY'. To construct the system Aj^ when A^^^ is known we make use of the theorem of § 104. We must therefore begin by constructing a system 5fc_i possess- ing the following properties : (i) it contains the form {ahf^ a^^'-'^h^''-^, (ii) it is relatively complete for the modulus (a6)^+^ Then the system derived from ^^-i and Bj^_i by transvection will be finite and complete with respect to the modulus (a6)^+^ and as it obviously contains f which is contained in ^^-i it is the system A^ required. 109. Accordingly we have now to shew how to construct the system Bj^^. . There are three cases. ?i I. If 2A?<- then any. form derived by convolution from a power of H^. = {ab)'^ a^^'-'^hx'^-^'' is of grade {1k+l) at least and therefore of grade {2k + 2) since all symbols are now equi- valent. Hence Hj^ is itself relatively complete for the modulus (a6)^+'^ and in this case th« system B^ consists of the single form {ahf' a^-'^h^-'^. (§ 74.) II. If 2A; > ^ then H^ = (ab)* a^^^^^h^'^-^ is of order less than n, say m. Now we suppose that the complete system of co variants for a form of order < w is known and we derive a system from Hk on the model of the complete system of a^"* as explained in i 79, 80. 120 THE ALGEBRA OF INVARIANTS [CH. VI Neglecting terms containing (a6)'*+2 ^q gan replace each copied form by a single term ; the system so derived is complete for the modulus (a6)^+^ and is therefore the system Bj^^i required. ft III. If 2k = ^ — a case which can only arise when w is a multiple of 4 — we have a rather different state of things. Here the form Hj^ = {ah)*ax^~^hx^~-^ is relatively complete for the two moduli (a6)^+S {ahY' {hc)"^ {caf\ the latter being an invariant J, and hence by Cor. II. § 107 the system derived by trans vection from A]c_y^ and ^j._i is relatively complete for the moduli (a6)^+^ and J\ calling this system G^ for a moment we have G, = F(C,) + J.P„ mod (abr^^ where Pi is a covariant of degree less than G^.. Further since Pj can be derived by convolution from / which is contained in G^, we have P, = F, (Cfc) + /. P^, mod {ahf+\ where Pa is a covariant of degree less than Pj. Proceeding in this way we see that Gk is a rational integral function of / and the forms in Gk together with terms involving the factor {ah)'*^^ Hence if we add J to the system G^ and call the total system Ajc it follows at once that A^ is relatively complete for the modulus Therefore in every case, given the complete system mod (a6)^ we can construct that mod(a6)'*+^; but the system A^ is/, thence we find the system A^, then from that the system A^ and so on, in fact we can construct the system A^ relatively complete for the modulus (a6)^+='. 110. Consider now a little more closely what happens when we come to the end of the sequence of moduli {aby, {oh)*, {aby ..., and first let n be even and equal to Ig. Then the system Ag^^ is relatively complete for the modulus {aby^, and the system Bg^i consists of the single invariant (ab)^ so that it is of course absolutely complete. 100-111] gordan's theorem 121 Hence the system derived from Ag^i and Bg_i by transvection is absolutely complete and it contains/, therefore it is the complete system of invariants and covariants ; further since Bg^i consists of a single invariant the complete system Ag consists of Ag_i and that invariant {aby^. Secondly let n be odd and equal to 2g + l, then the system Ag_j^ can be constructed and it both contains /and is relatively complete for the modulus {ahy^. The system J5^_i is derived from the quadratic {aby^ a^bx by the same convolutions as the complete system of the quadratic oi^ = ^^ is found from this form. This complete system being a^' and {ci^y the system Bg^^ consists of {abyo aj}^, {abya {ac) (bd) (cdy^. This system is relatively complete for the modulus {ahy^'^^ by § 109 II, and this being a vanishing invariant it follows that jB^_i is absolutely complete. Hence the system derived from Ag_^ and Bg^^ contains / and is absolutely complete, that is it constitutes the complete system of/. To recapitulate — the complete system mod {aby can be written down at once, then from that we deduce the complete system mod {ahy and proceeding step by step we can finally construct an absolutely complete system as the last step in our series. We have therefore proved that the complete system is finite, for all the systems A^, A^, ... are finite, and we have shewn how to construct it on the assumption that the systems for forms of lower orders are known — the proof is thus inductive in its nature. 111. We shall illustrate the above process by applying it to the quadratic, cubic, and quartic. (i) Quadratic. The system Aq is and the system B^ is {aby, hence the complete system is a^\ {aby. 122 THE ALGEBRA OF INVARIANTS [CH. VI (ii) Cubic. Here Aq is fz=.ai = hi = etc. and 5i is (abf a^h^, {aVf {ac) (bd) {cdf, in fact H, {H,H)\ This system B^ is absolutely complete, therefore the system derived by transvection is the complete system. It consists of f,H,(H,Hy = A and {f'^,m)y. Proceeding as in § 88 we can shew that the only irreducible transvectant is (/, H). (iii) Quartic. Here A^ is /=««' = &«;'.•., £o is H = (aby a^bi, and this is complete modd (aby and (aby (bey (cay. The system derived by transvection is If 7 > 2 this has a term containing the factor (aby (acy which is congruent to zero modd (aby, (aby (bey (cay. Hence we need take only 7 = 1 and thence only a = l, /3 = 1, and we find that / H. (f, H) is relatively complete modd (aby, (aby (bey (cay. Therefore / H, (f, H), (aby (bey (cay is complete mod (aby and is the system Ai. Then Bi being the invariant i = (aby we have for the complete system f,H,t = (f, H), i = (aby, j = (aby (bey (cay. 112. We shall now apply the principles of §§ 73, 76 to the n deduction of a complete system mod (aby for the binary form of order n. The system ^o consists of /= a,;" = fta;" = etc. and £o of H = (aby a^-^ fca,"-^ .... 111-112] gordan's theorem 123 The system Ai is derived by transvection from Aq and Bq. Now (/-, H^)y has a term containing the factor {aVf{acf if 7 > 1, and since such a term is = mod {ahy the transvectant may be rejected. If 7 = 1 the transvectant contains reducible terms unless a = yS = 1, and hence A-^ consists of f,H,{f,H) = t The system Bi is (ahy a^-^ h^-^ and -4.2 is derived by transvection from A^ and B^. If the index of a transvectant be greater than two it contains a term having a factor {ahf {acf and this is = 0mod(a6)«. (§70.) We need only consider the cases in which the index is ^ 2, and since the order of each form in A^ is certainly greater than 2 (in fact ^ ^ 4), products of forms may be rejected. There remain transvectants of each form of Ai taken simply with H^ = {ahYa^''-'h^'^. For the future we shall only write down the determinantal factors of a covariant. » Transvectants with / give rise to {ahy {he), {ahy {hey. Those with H give {ahy {he) {cdy, {ahy {hey {cdy, and finally those with t give {ahy {be) {edy {dc), {ahy {bey (cdy (dc). Now by § 76 (ahy (bey (cdy = (ahy (cdy, mod (ahy ; hence (ahy (hey (cdy (de) being a term of {{aby (bey (edy, e^"} 124 THE ALGEBRA OF INVARIANTS [CH. VI we have (aby (bcf (cdy (de) = {(aby {cdy, e^% mod {aby, for all expressions derived by convolution from {aby {bey {cdy are ' =0 mod {aby. (§ 73.) Now a term of the last trans vectant is {cdy . {aby {ae), .-. {aby {bey {cdy {de) = {cdy . {aby {ae), mod {aby and accordingly may be rejected. Finally {aby {be) {cdy {de) is reducible as being the Jacobian of a Jacobian, and the system A^ consists of f, {aby, {aby {be), {aby, {aby {be), {aby {bey, {aby (be) {cdy. 113. Before proceeding further we shall develope the results of § 76 by shewing that a symbolical product T containing the factor (a6)^ {bey {cdy, in which X is even and equal to fi + v, can in general be expressed in terms of covariants that are either reducible or of grade greater than X. The above reduction of ^{aby{bey{cdy{de) is a case in point. In fact r is a term of {{aby {be)'' {cdy, 4>Y which we write {T, jty. Hence V = {T, j>y -^X{f,^y, p<p = {T, 4>y + 2 {T, 4>y', mod (a6)^+\ since T derived by convolution from {ahY {bey {caf is of grade greater than \, § 73. Again T={aby.{cdy-\-C^^, (§76), r = [{aby . {cdy, <f>]f> + {{aby . {cd)\ ^j"' + Cx+x. 112-114] GORDANS THEOREM 125 Now if 2n — 2X^p each of these transvectants contains terms having {cdy Cx"^'^ dx^~^ as a factor. Hence {{ahY.{cdY,^Y = (cdy {{ahf, <t>Y + 2 {{aby . (cdf, 0}', mod (abY+\ a < p, and by continuation of this process we can express T entirely in terms of reducible covariants and covariants of grade greater than X; it suffices to remark that the index a diminishes at every step. It is quite easy to see that the condition 2n-2\^p is satisfied in all our cases — at any rate it will be in the course of the subsequent work. 114. Returning now to the general form, B2 consists of Hs^iabyaaT-'ba;''-', and A3 is derived by transvection from A2 and B2. The argument used in evolving A^ and A.2 enables us to see (i) that transvectants with index > 3 may be rejected, (ii) thence that transvectants of products or powers of forms may be likewise rejected. We are therefore left with transvectants of the forms of -4 2 taken simply with H^, the index being :}> 3. Omitting Jacobians of Jacobians and forms having a factor {aby (bey (cdy, where /j, + v^6, we have from/ (aby (be), (aby (bey, (aby (bey; H, (aby (be) (cdy, (aby (bey (cdy, (aby (bey (cdy; (aby (be), (aby (bey (cdy (de), (aby (bey (cdy (de) \ (aby, (aby (be) (cdy; (aby (be), none ; (aby (bey, {aby (be) (cdy (dey ; (aby (be) (cdy, none. 126 THE ALGEBRA OF INVARIANTS [CH. VI Hence we have found for A3 the above ten forms in addition to those of A2. Putting aside the question as to whether any of these ten new forms are reducible, a continued repetition of the above process establishes the fact that all the forms of the system Aj^, relatively complete for the modulus are included in the set (abf {her (cdf {dey {eff" {fgY" ...» where the exponents satisfy the followipg conditions : (i) ^i^^h (ii) \, \', \", . . . are all even, (iii) \>X' + /M, V > \" + fjf, . . ., (iv) no two of the exponents /*, fi', ... are equal to unity. In fact (ii) follows immediately from the way in which the covariants are formed. (iii) results from the application of §§ 73, 76. (iv) is the expression of the fact that the Jacobian of a Jacobian is reducible. Ex. (i). If the orders to, n, p of the forms /, (f), yl^, be each greater than two, then Ex. (ii). For a form whose order is greater than four the covariants {abf (be) {cd), {ab) (bcf (cd), {ahf {bcf {cd) all vanish identically. Ex. (iii). If « > 5, then {bcf {caf {ahf aj"-* bx^-^c^"-* = (ab)* {acf a«»-6 Jx""* Cx"-2 - \ {abf a^^-s bj'-^ . Cx\ Ex. (iv). If » > 4, then and if w > 5, * Cf. Jordan, Liouville's Journal, 1876, 1879. 114] gordan's theorem 127 Ex. (v). For a form whose order is greater than three prove that Hence replacing (/, /)* by i express (H, 3)^ as a linear combination of Si, fii, /)',/'{/,/)' and finally express t^ in terms of the irreducible forms of the system. Ex. (vi). Prove that all irreducible covariants of degree four and rank not greater than - are included in {abf^ (bcf {cdf 71 where 2X :^ - and X > /i > v. Ex. (vii). In § 103 if no A be of order greater than m and no B be of order greater than ?i, then no form of the system C is of order greater than m + n — 2. CHAPTER VII. THE QUINTIC. 115. To obtain the complete irreducible system of covariants of the quintic, we follow step by step Gordan's proof of the finiteness. Let us briefly recapitulate. The complete system of forms, which are not expressible in terms of covariants having a symbolical factor {ahy, is first found; this is called Aq, it is the complete system mod {aby. Generally A^ is used to denote the complete system mod (a&)^+^ To obtain the system J-^+i from the system A]c, a subsidiary system of forms Bjc is used. This system is a system of forms having ^ = (aft)^"*"^ a^^'^^'^^ hx^~^~* for ground-form. When the order of this form is less than n, Bk consists of its complete irreducible system. Otherwise if the order of (ft is greater than n we may take for B^ the single form <^ ; while when the order of <}> is equal to n, the system Bk consists of <f) and the n n n invariant (abf {bey {caf. Then it has been proved that the system A^+y^ may be obtained by taking transvectants of products and powers of forms from A^ with products and powers of forms from B^. 116. The quintic will be written f=<^i = bx'= The system A^ contains /only. The system B^ contains only {abya^bi = H. The system A^ is then obtained from the transvectants 115-116] THE QUINTIC 129 If 7 > 2, this transvectant contains a term having a factor {hcf ; such a term can be expressed as a sum of symbolical products each containing a factor {aby, and is therefore = 0, mod {ahy. Hence we may reject these transvectants when 7 > 2, for all trans vectants which contain reducible terms may be rejected. The transvectant {H,fY contains the term {ahf {hcf aAcx' = U- (a^)' «x&a; (§ 51, Ex. (vi)). The system ^^ then consists of / H, if, H) = t. The system By is built up from the form {aby a^ba: = i, this is of order < 5, hence we must take the complete irreducible system of the quadratic i. The system By^ then consists of i, {i, iy = A. The system A^ is now the complete system of forms for the quintic, it is made up of the transvectants U={f<^HHy, i^A'Y- Since A is an invariant we may suppose that e = (if at the same time we remember that A belongs to the complete system). Since H is a, form of even order, and i is a quadratic, all trans- vectants are reducible except those which have (i) /8 = 0, (ii) a = 0, 7 = 0. y3 = 1. Again t is the Jacobian of /and H, therefore ^^=-M(/./)^ff^-2(/, Hyf.H+{H, Hy.p] s-^jy^'mod {ahy. (§ 78.) Hence any transvectant F, in which 7 > 1, can be expressed in terms of transvectants V in which the degree of the product on the left has been decreased and that on the right has been increased together with reducible terms (| 105, Cor.): for as we have just seen if /3 > 1 then U is reducible. G. & Y. 9 130 THE ALGEBRA OF INVARIANTS [CH. VII Accordingly we have the following cases to consider: (i) a=l or 2, 13 = 0, y = 0, (ii) a = 0, /3 = 0, 7=1. (iii) a=l, yS = 0, 7=1, (iv) a = 0, /3=1, 7 = 0. All other transvectants are reducible or are expressible in terms of these. (i) a=l, 2, /3 = 0, 7 = 0. The irreducible transvectants are (/ 0, (/ *)^ (/ ^o^ (/ i'r, (/ ^J, (/^ ^T- To see that the other possibilities contain reducible terms we shall take one example. The transvectant (/^ i^y contains the term (/, i^ (/, i'Y. (ii) a = 0, ^ = 0, 7 = 1. Since t = - (aby (be) a^h^c^^ and (6c) (6i) c^i^ = \ {Q)cf ij" + {hif d - {cif K% the term — {ahy (be) (bi) a^ b^Cx^ix of (t, i) is reducible. Similarly the transvectants {t, ij, (t, i*y, (t, i*y contain reducible terms. Thus (t, i*y contains the term - (aby (be) (ai'y {ai") (bi") (ci"y (bi) c^H^ which is at once reduced by means of the above identity. We are left with the forms (t, iy, (t, iy, (t, ij, (t, i*y, (t, i^y. (iii) a=l, ^ = 0, 7 = 1. The only irreducible transvectant here, is (f.t,iy\ To see that the other transvectants are reducible it is sufficient to' remark that for example (/. t, i^)'* contains the term (/, ij.(t, iT / / ... 116] THE QUINTIC (iv) a = 0, ^ = 1, 7 = 0. The trans vectants {H, i), {H, if, (H, i% (H, i% {H, t^/, (H, i^y prove to be all irreducible. We are left with 23 forms which are as follows* : 131 Degree Order 1 2 3 4 5 6 7 9 1 / 2 i H 3 1 ihff ih f) t 4 A ihff)' {i,H) 5 {i\f)' (iM? {i, ty 6 (S\Uf {z^Hf 7 {i\ff {i\ ty 8 {i\ Hf ii\Hf 9 {p, tr 11 {i\tf 12 (,•5, /2)10 13 ii', tf 18 ii'Jtf' * One very obvious remark is to be made regarding this, and all other complete systems obtained by the present methods. We are assured that every covariant can be expressed rationally and integrally in terms of those retained in the complete system, but there is nothing in the process to shew that the latter are all irreducible, except in so far as failure to reduce them may be taken as evidence in this direction. Theoretically then Gordan's process gives an upper limit to the irreducible system. The enumerative method, depending on the generating function, introduced by Cayley and finally developed by Sylvester and Franklin (Am. Jour. vol. vii.) gives a lower limit to the system and when the two methods give the same result the irreducible set has been obtained. The results even when identical have to be received with some caution on account of the enormous labour involved. 9—2 132 THE ALGEBRA OF INVARIANTS [CH. VII 117. It is found that for discussing the properties of co- variants, it is convenient to have the indices of the transvectants which express them as low as possible. On this account it is usual to replace some of the forms in the irreducible system just given by others which differ from them by reducible terms only. In the first place the covariant is of fundamental importance in the quintic system. It is a cubic, and the system of forms for which it is a ground-form are irreducible when considered as forms belonging to the quintic. Now (H, i^y contains a term (aby(a{)^hiya^h = (j,jy, . accordingly we shall take the irreducible form of degree 6* and order 2 to be (i. jy = 'r- Similarly the form (t, i^Y may be replaced by {j, r), for (t, i^y contains a term (aby (be) (aiy (bi'y {cij a^ c^ = {{aby{aiy{bi'yaj>^, (c{"yc,') = (j, t). And (f^, i^y^ may be replaced by the invariant of j, (r, ry. This invariant will be denoted by C, the proof that it may be included in the system instead of (/^, i^y^ will be given later (§ 121). It will be found useful to denote the term (aiy (aiy (biy {bi'J (ai'O (6t'0 of this transvectant by M. Then M may be taken as the invariant of degree 12. It may be recalled in fact in connection with the simultaneous system of a cubic and quartic (Gundelfinger, Math. Ann. Bd. iv.) that the two results originally agreed, but a revision of the generating function led to a reduction of the lower limit which it theoretically gives, and afterwards two forms included in the irreducible sj'stem as derived by the methods of Gordan and Clebsch were found to be reducible. The complete systems for the binary forms up to the octavic may be considered as accurately determined by the two methods combined. 117-119] THE QUINTIC 133 Besides j and r, there is one more quadratic covariant, given in the list as (H, i^f. This is equal to {{H, iy, i); we may- substitute T for {H, t^y, and hence take as the remaining quadratic covariant {r,i) = -% 118. The linear covariants. if, i^y = {aif {aHy a, = - ( j, if = a, (/, i'f = (ai)^ {aif {aif') 4" = (a, i) = - ^. {t, i*y contains the term {aby (be) {aiy {hij {ci'J {ci"J a^ = {{ahy{aiy(hi'ya^h^,a) = {r, a) = 7. (t, i'^y contains the term (aby (be) (aiy (bij {ai") {cif'J {ci^^'y i^' = ({aby (aiy (biJ (ai") ij'b^, a) - ^ (aby (aiy (biJ (ai") (bi") . a, of which the second term is reducible and the first = ((T,i),a) = -(^, a) = -S*. 119. The invariants. (i,iy = A, {H,i'y = ((H,i%iy. The latter may be replaced by (r,iy=B. (f'\ i^y has been replaced already by M. (ft, i^y^ contains a term ((/ i% (t, *■«)"). and hence may be replaced by (a, B) = — R. Taking the Jacobians of the 6 linear forms two and two, we obtain the 6 invariants (a^), (^7), (7a.). («s), (m> (75). * This is the definition of the linear covariant 5 of degree 13 given by Clebsch. In Gordan's book 5 = (r, ^)=-{S-, a) -^(i, t)'^ . a. In other respects the letters common to the two books are identical in meaning. 134 THE ALGEBRA OF INVARIANTS [CH. VII The values of these invariants are* (a/3) = - (aiy {aij {ai") {WJ {h^^f {hi") = -M=- {{a)\ (/S7) = (id) (ra) (ir) = (^a)^ = (Ba) = R, {ya) = {ray = ]Sr, {aB) = -R. Now 8 = (({, t), a) = (it) (ra) 4 + i (^T)^ ct, , hence (^8) = {ia) (W) {i'r) (ra) + ^B . {iaf (78) = ((Ta)T^, {iT'){ia)r^-kB,a) = \{CM-BN). Also N = (7a) = {ray = {jif (j'iy (rj) (t/) = (>■) U'iyiv) {(ri) (jj) + (ri) (fi)}. Now from the theory of the cubic we know that any symbolical product which contains a factor (rjY is zero. Hence j^=(j.r)(ji)(Fnrj){Ti) = i(i?") (ri) {(ji) U'ij (Tj) - (fi) (jiy (t/)1 = ^(i?"') (ri) {(jf) (*•*") (/O (rj) + (ji') (j'i) (fj) (ri')} = Hjjj (V) (rj) . (iij - i ((jjyj.j.', (ri) (ri') ij^'f = IAG- i(T. I [(tO^ . tV^ + (rt')^ . i.' - (Hy r,^]y = i^(AC-R). 120. The third transvectant of / with j is identically zero. For (/ jf = - («x^ (biy Kj = - (aby {uy aj^ = + 1 (aby {(ai)' bj> - (biy a^] = \ (aby 4 . {(ai) b^ + (bi) a^} = (i,i) = 0. This property is sufficient to define j. For if yfr be an arbitrary cubic then (/, yfry is a quadratic. And in order that (/, yjry may * In future when no confusion can arise the comma between the two forms in a transvectant will be omitted. This is the usual practice, cf. Stroh. 119-121] THE QUINTIC 135 vanish identically the coefficients of x-^, XjX^, x^ must be separately zero; giving three equations to determine the ratios of the four coefficients of i/r, — see Chap. xii. Again (/T)^=(«i)(«/)0}7«.* = - 2 {ajf (ajy a^j^jj, since {ajf a^ = = - 2 {ahj {acf (bif {cij aj)^c^. Now iahf {i'hy (cby =0 (§77). (aiy {hif A (ci) (acf {bey (i'cy (bay (i'ay (caf Hence, if S include all possible expressions obtained by inter- changing a, b, c, X(aby(biy(icy(cay = (aby (ciy (ci'y + (bey (aif (aij + (cay (uy (bi'y + 2A.(ahy(bcy(cay. And therefore (/ r)'^ = - 2 (aby (acy (biy (ci'y a^b^c^ = - (aby aj}^ . (aiy (ci'y c^-^A. (aby (bdy (cay a^b^c^. But (aby(bcy(cayaa,b^C:, = (aby (bey (ca) [ — (be) a^ — (ab) c^ a^Cx = - (ahy (bey (ea) a^c^ - (aby (bey (ca) a^e^ = \ (bey (ea) (ab) aj^ + \ (aby (be) (ea) ei = -(ify=j- Therefore (f,Ty = -i.a-lA.j. 121. To obtain the relation between the invariants C and M, we take the expression for M and introduce in it so far as possible symbols referring to the cubic j and its Hessian t, for G=(Try. 136 THE ALGEBRA OF INVARIANTS [CH. VII Now M = (iaf = {3ir{j'iJ{F){j'i") = (» im [{ja) (ih) - ijb) {ia)Y [(/a) (i'h) - (j'b) (i'a)f = (» (/&) [ - 2 (ja) (jb) (ia) iib) + ijby {ian [{j'ay (i'bf -2(fa)(fb){i'a)(i'b)l since ( jaf ai = 0, and (/ft)* b^ = 0. In this expression we will introduce, as far as possible, symbols referring to ; ; for {iaya^^=j = {i'byb^\ M is then seen to be the sum of four terms, viz. W{j'h){jbnianj'af(i'bf = ijj") UT) UfJ UJ'J = - (tt)'' = - C : 4 {jay ij'by (jb) (fa) (ia) (i'b) (i'a) (ib) = 2 (jay (j'by (jb) (j'a) [(iay (i'by + (iby (i'ay - (uy (aby} = ^(jj'J(jj"J(jj"'){j'j") - 2A . (jay (j'by (aby [(ja) (j'b) - (jj) (ab)] = - 4C + ^ . (jjj (aby {(ja) (fb) + (jb) (j'a)], since (jay ai = 0, = - 40+ 2A . (rt7= - 4(7 + 2AB : - 2 (jay (jay (jb) (j'b) (i'by (ia) (ib) = 2 (jay (fay (jf) (ff) (ia) (if) = (i«) (fa) (ia) dj") [(jay (ffj + (fay (jj'J - (fay (jfy] = - (ray (fay (ia) (if) = - (ra) (rf) (fay (iay = C : the last term of M - 2 (jby (j'by (ja) (fa) (iay (i'a) (i'b) is obtained from the one just considered by interchanging i and i', a and b, its value is therefore C. Hence M = -G - 4>G + 2AB + 2G = -SG+2AB. 121-122] THE QUINTIC 137 122. The covariants of orders 0, 1, 2, with the exception of i, have been replaced by transvectants, of index not greater than 2, of simpler forms. We may simplify the expressions of the others in the same way, in fact this has been done already for two of the covariants degree 3, viz. j and (j, r). The remaining cubic covariant (/, i^f = ((/" iy, i) The transvectant (H, if may be replaced by its terra it will be convenient to write this covariant The remaining quartic covariant is and may then be replaced by (p, i). Now since (ajf a^ = 0, {aJT o,xjy = (oy? CLxjxay = i {{ajy ai^y + 3 {a^y a^ix(^y\ Hence {p, i) = {ajf (ji) a^% = {ajf (ai) aij^i^. Now ' (/,«)=(/-(>?>) = - (aj) {(«*)> - (aj) 4}' O'i = 2(aj)2(ai)4ja,a«^ = 2 {p, i), which gives another expression for the same covariant. For order 5 we have only to consider the transvectant {t, i^y ; this has a term (ahy (be) (dy {aij a^b^'cj", hence this transvectant may be replaced by {{H, iy, {/, iy), and therefore by (P'j)- Lastly the covariant order 7 may be replaced by the Jacobian 138 THE ALGEBRA OF INVARIANTS [CH. VII 123. To express any transvectant of two covariants of the quintic in terms of members of the irreducible system, it is in general advisable to use as far as possible symbolical letters referring to covariants such as j, i, t, ^ etc. instead of those belonging to the quintic itself. We give here the values of some of the trans vectants, partly for the sake of reference, and partly as examples; we would recommend the student to verify a few of them*. The following table gives the 2nd transvectants and Jacobians of the quadratic and cubic covariants. 2nd Trans- vectants Jacobians i r ^ J U, U, r) i A 5 ^{Bi-Ar) -U,i) lai-¥\Aj ^ar + \Bj T B G hiCi-Br) -U,r) \ar + \Bj hCj s J}^ — \aT l^^+hi hr J — a -y T -h^T-ya -h" u, *■) i/3 -§y i«-i^a -i^ -J{2a2-fylr-3i5i} -i{3yJ + a(J,r)} 1 y -iCa IBr + iCi \Ct The third transvectants of the cubic covariants are (j, ( j, i)f = B, (j, (j, T)y = G, {{j, i), (j, r)y = 0. In obtaining the values of Jacobians it is well to remember the formula m — n proved, Chap. iv. § 77 ; where /, <f> are binary forms of orders m, n respectively ; and the order of each of the forms /, ^, i/r is not less than 2. This will be of frequent assistance, since of the 15 • Should he experience any great difficulty he will find some of them worked out in Gordan's Invariantentheorie and Clebsch's Bindren Fortnen. 123] THE QUINTIC 139 «o 1 + 1 i 1 1 1 s 1 r 1 + + Hw 1 1 s Hn + oq r-te 1 H« + 1- f ■* + t r^« HN 1 1 05 ?^ + 1 1 1 H<N 5 1 "6 Hn + 8 He + cs H<N o; H» 1 He + «1 Kin 1 > + 1 B 1 Him 1 I 1 HN 1 HN J HN + 1 1 HN + B '^ HN 1 •<^ H« 1 05 HN 1 He + Hm 1 HM e <«. ?«• «o 1 1 HN 1 1 HN HM 1 + 1 HM 1 + 8 HN 1 B "^ •- (- Sf} -^ a. •5 140 THE ALGEBRA OF INVARIANTS [CH. VII irreducible covariants of the quintic for which the order is not less than 2, 9 have been expressed as Jacobians. It is useful to know the values of the following transvectants 124. For purposes to be presently explained the transvectants (/. cf^ (/. '^^y, ••• will be required. Such transvectants are calculated step by step, first (/, a), then (/, a*)'' = ((/, a), a), and so on. • It will be useful then to know the values of the transvectants of certain of the covariants with a, f3, y, 8; when these are known the values of such transvectants as (/, a')' may be obtained with great ease. 125. Syzygies. It has been proved (§ 77) that the product of two Jacobians, or the square of a Jacobian can be expressed as a sum of terms, each term being the product of at least three forms. Now nine of the covariants of the quintic are Jacobians, — we must exclude the forms (la), (ra), (^a), (a/3), (aS) for one at least of the quantics in each of these Jacobians is of order less than 2 : hence we have 45 syzygies. A more general method of obtaining syzygies is given by Stroh. He considers four different forms, and seeks to obtain the syzygies which are of unit degree in each of these forms. First for three forms there is evidently only one such relation, that for weight unity /i (AA) +Mfs,A) +/s(A,A) = 0, this is written for short { f 1/2/3) = 0. The other syzygies obtained by Stroh arise from the symbolical relation (ab) Cxdx 4- {cd) axhg. = {ad) hxCx + (c6) axdx , which may easily be verified. Raise both sides of this identity to the power i and expand by the binomial theorem : hence S iV) {abf(cdy-^aJ-^bx^-^Cx'dx' = 2 (l) {ady {cby-^aj-^dj-^cx'^bx- 123-125] THE QUINTIC 141 Hence This is written Other syzygies may be obtained from it by interchange of the various quantics concerned. But it will be noticed that (fif 2/3/4)% is unaltered if one pair of letters is interchanged and at the same time the other pair. Also this expression is only changed in sign i( / and / are interchanged. Hence only three distinct syzygies of weight i are obtained in this manner. If z = 2 (////). ^ (//f//+(//r// - {//)%/ - (//y// Whence {////}. ^ i [(////% + (////). + {////%] = (//) (//) + (//) (AA) + (/a/4) (//s) = 0, a result already well known. Also [/A//1 = i [(////% - (//A/% - (////a).] • = (//T// + (/3/yy./ - (//)%/ - i//y// + 2 (//a) (//) = 0. This is the same relation as that given (§ 78) for the product of two Jacobians. It will be seen at once that the other syzygy of weight 2 is deducible from this. The syzygies of higher weight may often be simplified, in the same way. The syzygy {///a/.}.= is remarkable from the fact that the forms in it need not be of order higher than unity ; while in the syzygy from which it was deduced, each of the forms is of necessity of order 2 at least. 142 THE ALGEBRA OF INVARIANTS [CH. VII In general in the syzygy (/iA/s/.)i=o each of the forms/1,/2,/3,/4 must be of order i at least ; but from these syzygies others may be deduced which are true for forms of lower order. Stroh, in his papers on syzygies*, deduces many such. We will obtain such a syzygy from that of weight 3, (/l/./3/4)3=0. If \, fi, V be three quantities whose sum is zero, we know that hence 3 {ah) (be) (ca) aj)^c^ = {hcf a^" + {caf b^^ + {abf c^\ We thus obtain h [(a^'h'c^'d^% - {(ix'cMz'b^% + {a^'d^'b^'c^^),] - (ab) (be) (ca) aJ)xCxd^ + (b.\ cj) (a^\ d^J + (c^^ a^') (b^\ d^J + (a^\ b^') (c/, d^'Y = 0. It will be seen at once that a^, b^, Cx, dx are factors of this syzygy, we may then divide by any one or all of these factors ; or else we may multiply the syzygy by a power of any one of them. The syzygy is then true whenever the order of each form is gi-eater than unity. It is not difficult to see that this derived syzygy is just as general as the original one in other words, the original syzygy might be derived from it. Stroh writes the syzygy just obtained in the form {fHq% = ( fqf iH) - (fqJ i<f>q) - (f<f>y (qq) +A(qq') <!>)' = o, where q, q' are quadratics, and /, (f) are forms of order 2 at least. Again from the syzygy * Math. Ann. Bd. 33, pp. 61-107 (§§ 18-22) ; Bd. 84, pp. 306-320, 354-370 ; Bd. 36, pp. 262-288 ; in § 3 of this latter paper he gives a list of syzygies of the kind just mentioned which he has deduced. 125-126] THE QUINTIC 143 we obtain {ffH\ =p i<}>cf>y + 6 (fff i<fy<f>y + ^^ (f/y - 2f<f> (fcf>y + 8 (f<f>y (/</,) - 6 [{f<i>yY = o. In this write i- for (f), where i is a quadratic. Now (i\i:^y=i[(nyy, (i\i^y=i(ny.i% 2(f,ijif,i^) = 2i{(f,iy,i).(f,{) —see § 78. Hence substituting these values in the syzygy obtained by writing i^ for <f), and then dividing the result by 2i^, we obtain =f{py - % {{fiy/y - [{fiyj + (n)-^ {f/y + ^"^ {ffy = o. From this may be obtained a syzygy by means of the operator (-v/r >j. This operator requires that i/r and / should be of the same order, but when the syzygy is written symbolically it will be seen at once that factors of the form ax and h^ may be introduced so that the relation is true whatever be the orders of / and i/r, provided that neither is less than 4. In the same way we may obtain a syzygy where both i and r are quadratics. 126. Application to the quintic. The forms of the quintic consist of /, H, i, j, r, p, a, /3, 7, B and some of the Jacobians of these forms. A large number of syzygies may be at once obtained by writing for the forms in the general syzygies of the last paragi-aph covariants of y. 144 THE ALGEBRA OF INVARIANTS [CH. VII We shall content ourselves with a few examples. {fiH)=f.{iH) + H{fi)-it = 0. (fjH)=f.(jH)-H(jf)-jt = 0. [fM^ = i(M' + Hi-^ + 2A>- +PA = 0. [ijri]^ = B (ji) - A (jt) - a^ + iy = 0. (ppii)i = -Bp + fm^ - ^Air - ^Bi^ -t'+^AJ(z= 0. {a^ryB}^ = \M{BN - CM) - \N{AN- BM) -B? = Q. From this last we deduce the relation connecting the irreducible invariants A, B, G, R. It is eas}^ to write down a great number of syzygies in this way. Stroh {Math. Ann. Bd. 34, pp. 354 — 370) has given a list of 168 syzygies of the quintic. The notation for the elementary syzygies given here is not quite the same as that used in Stroh's paper, but the notation is there explained ; the notation here has been mainly taken from a later paper by the same author (Stroh, Math. Ann. Bd. 36, pp. 262—288). 127. Reducibility of syzygies. If be any syzygies, and if Pj, Pg. ••• he any products of forms such that Pi^S^i, Po'S'a, •.. are expressions all of the same degree and order, then is a syzygy. In this way it will be seen that an infinite number of syzygies may be built up. ^ syzygy 8 = is said to be reducible, when /S=P,5fx + PA+-., where 8^ = 0, 8^=0, ... are syzygies whose degree is less than the degree of *S = and the P's are covariants. Otherwise it is said to be irreducible. It will be seen at once that any syzygy which contains a product of only two irreducible forms must be irreducible. All the irreducible syzygies which have yet been found for the quintic are of this nature. 126-129] THE QUINTIC 145 128. It may happen that certain syzygies ^, = 0, ^^2 = 0, ... Si = are such that certain products of forms Pj, P^, ... Pi may be found, for which P,8, + P,S, + ...+PiSi=0, the expressions P, S being regarded as functions of the con- comitants — which for the moment are treated as independent variables. Such a relation is called a syzygy of the second kind. The following is an example, (/1/./3/4) =/i (fJsA) -A (/1/3/4) +/3 (/1/./4) -A (AM) = 0. Thus for the quintic (fHvj) ^/(Hij) - H ifij) + i i/Hj) -j if Hi) ^ is a syzygy of the second kind. Syzygies of the second kind may clearly be reducible or irreducible. Between them may arise syzygies of the third kind, and so on. The following questions at once present themselves. ' Is the number of syzygies finite when the system of forms is finite?' ' When the syzygies of the first kind are finite in number, are also those of the second and of higher kinds finite V 'Is there any limit to the number of kinds of syzygies which arise from a finite system of forms ? ' All these questions have been answered in the affirmative by Hilbert {Math. Ann. Bd. 36, pp. 473—534). They are partly considered in Chapter IX. 129. The typical representation of the binary quintic. For special purposes, some particular linear transformation of a binary quantic may have peculiar advantages. Thus any particular term of the quantic may by a special transformation be made to vanish. If the quantic has two linear covariants, such that the determinant formed by their coefficients does not vanish, these may be taken for the variables : the transformed quantic will then possess the property that every one of its coefficients is an in- variant. We proceed to prove this. Let a^, ^^ be two linear G. & Y. 10 146 THE ALGEBRA OF INVARIANTS [CH. VII covariants of the quantic a^^, which are such that (ay8) is not zero. Then raising the identity (a^) a^ = (a/3) a^, - (aa) ^^ to the nth power, we obtain (a/3)« . a^« = (a^y . a^« - n {a/3y-' (aa) a««-^/8^ + . . . The expression on the right is the transformed quantic, and from the symbolical form of the coefficients, it follows that they are all invariants. For the general quintic any pair of linear covariants may be chosen ; for example those which we have written a and /3. The coefficients may be easily calculated with the help of the table given on p. 139 ; they are as follows : (/, a'^y = - UN- ^AM + \~^ 31 +\^ {2BM-AN) = (m. \A' {f,a^y = -AR(M+~^ {f,^y = -M' + ^AM{AN'-BM)-\-fi^. Further the invariant (a^) = — M = SC— 2AB must not be zero. To be more accurate the coefficients given above should be divided by (— My. In the expression for any covariant in terms of the actual coefficients, the above transformed coefficients may be substituted, the covariant multiplied by a power of the determinant of transformation is then equal to the expression thus obtained. In this way any covariant may be expressed in terms of the invariants and two of the linear covariants. To illustrate a different method of expressing any covariant in terms of the invariants and two linear covariants we shall 129-130] THE QUINTIC 147 obtain j in terms of the invariants and the covariants a. and 3. Raising the identity Wjx = (j8) ax - (» ^x to the third power _ R^j = (j, 8-^y a'-S (j, B^af a^S + 3 (j, Sa^f aB^ - {j, a'f S^. But by the method of § 124 {j,aB^) = -^NE, {j,hJ = -^R{GM-BN). Therefore B?j = ^{GM- BN) Q? - %Nci^B - B\ 130. Given two binary forms of the same order, in particular two quintics, can one be linearly transformed into the other, and if so how ? The reply (in part) to the first question is that if the absolute invariants of the two quantics are equal to one another, and if they each possess a corresponding pair of linear covariants of which the determinants do not vanish, then the quantics are transformable into each other. The question will be found dis- cussed in Clebsch, Bindren Formen, § 92 ; and for the case where there are no linear covariants in § 105. When two quintics have equal absolute invariants and one of the 6 invariants (a/S), (ay) ... is other than zero, say (a/3), we may transform one quintic into the other thus : — Let unaccented letters refer to one quintic, and accented letters to the other ; we transform each quintic, so that the variables in the first are a, /3, in the second are a, yS'. Thus f=AoaP + 5A,a*^ + f = Ao'<x'' + 5A,'a:'B' + A Let the ratio -r? = r, then since the absolute invariants for A the two quintics are equal it follows that A^~ A'^' and hence | = r=. 10—2 148 THE ALGEBRA OF INVARIANTS Similarly ^, = r3, and -^, = r^ Hence also <■ A ' ' ) A ' ' > A ' '» .n.Q Jlj M.2 :ttl-r~^ ^*-r-« ^-r~^ jO-s Ji-i Ji-s [CH. VII The quintic /' may now be transformed into / by means of the transformation 6 a = T^a', /3 = r^^^'. 131, Associated forms. If yi, y^ is a pair of variables cogredient with aj^.a;.^; then the two forms are invariantive. Now regard x for the moment as a constant, and the two equations just written down as equations of linear transformation to transform from the variables y^, y^ to new variables f, 77. The variables of the transformed form are co- variants, hence its coefficients are invariants — or to be more accurate co variants, for they contain a; but not y. Let us proceed exactly as in § 6. The determinant of transformation is (I'Z) = «."=/• From the identity we obtain the transformed quantic i^Y 6/ = {h7)Y |« - n (bvT-' (6f ) f«-^ 77 + (I). Let us calculate the coefficients of the transformed form for the case of the quintic. (bvY m' = hj^ iha) a,* {ha') aV = ^H .f, (brjy {b^y = b,^ (ba) {ba') iha") a,*a',*a\* = ^b^^ (ba) a^*a'^'a"a? {{haj a'V + (ba'y aV - {a'a'J b^'] = -t.f-^H. {ba) b,*a^* = -t.f, 130-131] THE QUINTIC 149 ihrj) {b^y = b^ (ba) (ba') {ba") {ba'") a^^a'^'a^^a'^* = {baj {ba) {ba") h^aja^*a"'^' . aV -^H .{ba){ba'")b^'a^'a'"^* = {bay {bay b,a7 aj J' - h H'f {b^y = (6a) {ba') {ba") {ba'") (6a»^) ax*aVaVa"Va'V = {ba'y {bay {ba}^) a'^'aj'a}\* .p + Etf. Now {{bayi^o:yb^aia'i,f) = j {bay {ba'y {{be) a^ali + 3 {ac) b^aiali + 3 {a'c) b^a^a'^^] c^* = {bay {ba'y {be) a^'a'a^c^" + ^{aby{ba'yaJ'a'^Kf = {bay {ba'y {be) ai a'^'c^* + f {aby {{a'b) a^ + {a'a) b^\ aV • / = {bay {bay {be) a^'aHe^' + f (/, i) ./. But the transvectant {{bay {bay b,aio!.\f) = {\ifj) = ^ (/ i) ./ Hence (6a)- {bay {ba}'') aia'^^e^'' = -{f,i)-f> and therefore {b^y = -{f,i).f + Htf. The transformation is then + o.[ir-lH^)^V' + {{f,i)f^-Ht)r,^ Now let O {y) be any covariant oi f{y), then when the above transformation is made, the coefficients of f{y) are replaced by the corresponding coefficients of the powers and products of ^, 7/ in the expression on the left. Let O thus transformed become ^' (^> "nX then ^ is equal to ^' divided by a power of / the determinant of transformation ; thus 150 THE ALGEBRA OF INVARIANTS [CH. YH This equation is an identity. We may replace in it y by ar ; when this is done, ^ becomes /and 17 becomes zero, hence - . , <!>'(/ 0) ^(^)= ^x ' Hence any covariant of the quintic is equal to its leading coefficient, when the original coefficients of the quintic are replaced by the corresponding coefficients in the form (I), divided by some power of f. From this we see that all covariants of the quintic may be expressed rationally in terms of the covariants fy H, x, t, (/, t), in such a way that / alone occurs in the denominator. Such a system of covariants in terms of which all covariants of a system may be algebraically expressed is called a system of associated forms. We have confined ourselves to the case of the quintic, the results obtained are however true in general. The coefficients of the transformed quantic may always be expressed as rational integral functions of/, the covariants of degree 2, and the Jacobians of these latter with / And this is in fact the simplest system of associated forms. The matter will be found fully discussed in Clebsch, Bindren Formen, ch. vii. The student who requires further information on the subject of typical representation will find it in the chapter just quoted and the two succeeding chapters of Clebsch's book. The reduction of the quintic to a sum of three fifth powers will be discussed in Ch. XI., and so nothing need be said on the subject here, especially since it concerns the non-symbolical treatment of the subject rather than the symbolical treatment. The special canonical forms to which the quintic may be reduced, when one or other of its invariants vanishes, will be found in Prof. Elliott's Algebra of Quantxcs. For a symbolical treatment of the subject the student is referred to Gordan's Invariantentheorie, or Clebsch, Bindren Formen, §§ 93—96. 132. The Sextic. The difficulty in obtaining the complete irreducible system of concomitants of a binary form increases very much with the order. The system for the sextic is obtained here ; 131-132] THE QUINTIC 151 it affords examples of a method of reduction applicable to forms of a higher order, but not required when dealing with the quintic. The arrangement in systems of forms whose grade does not exceed a certain number is followed as before. The system A^ contains only /; Bq contains only The system Aj consists of /, H, (/ H) = t. For Bi we must take the complete system of Now i is a quartic, and its complete system is To find the system A^ we must take the transvectants of powers and products of forms of A^ with powers and products of forms of i?i. Now the form (i/Y can be shewn to vanish, for (ify = (abY(bcy{ac)a^c,' = — l{ah) (be) (ca) [{aby (be) a^ci + (bcY (ca) b^aj^ ■^{cay{ab)cJ)J''\ = l{ab) (be) (ca) [(aby cj- + (bcY a^' + (caY 6^*], on using Stroh's series / / / ( / J / \ V 1 1 1 A But (aby (be) (ca) c^* = 0, since it changes sign when a and b are interchanged. Hence (i/Y = 0. The quadratic covariant (i/Y is of great importance ; it is usually denoted by the symbol I. If any covariant can be expressed as a symbolical product in which the factor (iay appears, it can be expressed as a sum of transvectants of I with other forms. For such a covariant = ((iay i^aj, 0)p + 2 ((iay aj, <!>')'> = s (I, <^y. 152 THE ALGEBRA OF INVARIANTS [CH. VII Again A = {i, if = {iaf (aby iA' + X (aby . i. And {{iay i^a^\ b^J = i {iaf bj> {{abf 4 + 3 {aby (ib) a^} = (iay (aby i^b^^ + 1 (iaf (aby b^\ but (iay i^ai = 0, hence (iay (aby i^b^^ = - 1 (iay (aby 6a,' = - f (l/Y- Now since a and b are equivalent symbols (iay (aby i^bx' = - ^ (aby ij" {(iay ba>^ + (ia) (ib) a^b^ + (iby aj] = - f (iay (aby i^^' + I (aby . i, and therefore ^ = i i{fy mod. (abf. Thus every form of Bi except i and the invariant (i, iy = modd. (iay, (aby. 133. We shall first find the system which is relatively complete with respect to the moduli (aiy and (aby (§ 107, Cor. II.). This is obtained by taking the transvectants of powers of i, with powers and products of forms of the system A^. Let us call this the system G. First consider the forms (i\P)y. We have (i,f), (i,fy- Every other one of these forms = mod. (iay. Next consider forms (i\ m)y. We retain only (i, H), for (i, Hy contains the term (iay(abyi^^a^'b^'; this can (§ 63) be linearly expressed in terms of covariants (iaya^^b^\ (ibyb^^a^\ (iay (ab) i^ax^b^\ (iby (ab) ij)^a^, (aby a^b;^ . 4*. Hence (t, Ey^Xi''^- id ./. The form {i, Hy contains the term (iaf (aby i^a^b^* ; 132-134] THE QUINTIC 153 and the form (i, H)* contains the term hence these may both be rejected. All other forms (i*, H^y contain a term having a factor {iHy. No one of the forms (i", ^)t need be retained, for if 7 = 1, the transvectant is the Jacobian of a Jacobian and another form and therefore reducible (§77). If 7 > 2, the transvectant always contains a term which involves a factor {ia)^ or a factor {idf, and therefore which = mod. {aif. If 7 = 2, the transvectant {i, ty contains the term = {{i,H)\f) mod. {iHY = \ (^^ /) mod. (aiy = \i {i, f) mod. {aiy. The general transvectant {i\ f^Hyfy may be treated in the same way. If e > 2, the transvectant contains a term which = mod. (aiy. And if 6 :}> 2, it is certainly reducible, except for the cases already discussed. The system C then contains the forms / H, t, i, a, ^•)^ (/ i), {H, i), if, iy. 134. To find the system A^ we must now take all possible transvectants of powers and products of the system C, with powers and products of the complete system of I. Now I is a quadratic, its complete system must then consist of I, {I, ly. The invariant {I, Vy is the same as an invariant already found, viz. : (t,A)^ = Hi Wy mod. (a6)« = \{liy mod. (a6)«. 154 THE ALGEBRA OF INVARIANTS [CH. VII Since Z is a quadratic and all the co variants of C are of even degree, we need only consider the transvectants of powers of I with each form separately. The forms a/), {i,fr=^\ {i\f)\{iM)\(i\fr,{i\fr are all irreducible ; so also is {I, H). The covariant (l, Hf contains the term = 2{(i,iy,fy = \ (iij {i'af i^'a,' + fi (i iyf. But the term is linearly expressible in terms of the covariants (§ 63) (nj ««,«, {iiJ (i'a) i^a^\ (ia)* a^ i^\ {iay{ii')i^^a^, {i'a)* a^H^*, each of which is reducible. Hence {I, Hf = \^ (nyf+ \li. Now, if )8 > 2 (^, Hy = ((I, Hy, i-^y-^ = (uy (D + x^ {H, ^-^/-^ hence these transvectants are all reducible. The covariant (I, t) is reducible, § 77. The covariant {I, ty contains the term i{i,sy,fy and is therefore reducible. The covariant, yS > 2, (/a, ty = {{I, ty, t-'f-^ and is reducible. Hence all the covariants {i\ ty are reducible. 134] THE QUINTIC 156 The covariants {I, i), {I, if, (^^ if are irreducible ; but {l\i)* = {l\{ahfa,'h^^f = ({l,fy, (IJffmod.iabf = 4 ((t, if, a iff mod. {ahf, which is reducible when considered as an invariant of the quartic i The form (l, (f, i)) is reducible, § 77 ; (I, {f, i)f contains the term • {{l,ff,i) = {^,i) = -v; (^^ {f, i)f contains the reducible term {l,{{l,f)M))- Similarly {l\{f,i)f, (l\(f,i)y may be reduced. The forms (z^(/,^•))^ (/^ (/,*■))«, (^^ (/,*•))« are however irreducible. The form (I, (/, if) is not reducible. Now (I, {/, iff contains the term (ii,ff,iy=2{A, if = ii(iif... (see § 89). Hence (Z", (/, iy^y is reducible when /3 > 2, for this = ^ (iif it-\ if-^ + X (Z«-i, l'y-\ ' Lastly (l, {H, i)) is reducible by § 77. {I, {H, i)f contains a term ((I, Hf, i) = X, (iif (/. i) + X, {li, i) and • {l'^,(H,i)y, /3 > 2, = (i-\(i,{H,i)fy which IS reducible. 156 THE ALGEBRA OF INVARIANTS [CH. VII Thus the system A.^ contains the forms of the system G together with /, (I, I)- and (i,n (ijy> {i^/y, (^^/)^ (^^/)^ (^^/)^ (I, H), (I, i), {I, if, {i\ If, (I, (/ ^))^ (/^ (/ i)y, {i\{f,i)r,{i\{f,i)r,{i,{f,in The system B^ contains only the invariant (/,fY', we merely add this to the system A2, and the result is the complete system for/ We append the following table giving the 26 irreducible concomitants of the sextic. Degree Order 2 4 6 8 10 12 1 / 2 if,/)' iff)*=i if,ff=s 3 (/, ir=i (/, ^7 if^l if, S) = t 4 {i, ly if If if I) {H,t) 5 ih I)' ii, I) (II, I) 6 {I, If ((/. if, I) ((/. i). If 7 if, i^Y if I'f 8 ii, I'r 9 iif,i),i')' 10 if, I'f if ^)* 12 iif,i),l')' • 15 Of, i), i¥ Ex. (i). Prove that if the covariant a, of a quintic /, is identically equal to zero, then also /3=0, 7 = 0, 8 = 0, 3 = 0, ipi) = 0, M=0, M=0, R=0, Bi = AT, {B-%A^)J=0, Bf+yr=0. Ex. (ii). In the Iqst example either j=0 or B = IA^. 184] THE QUINTIC 157 Prove that in the former case ever}- covariant vanishes with the excep- tion of and that these are connected by the relations At + \i\fi) = 0, 2[(/,^•)P + ^^•2+/2J=0. Ex. (iii). Prove that if o vanishes identically and^' is not zero, then B^IA\ t^lAi, A{Af^^ij)^0. The latter result gives an alternative, but if then Aj=^ (ij, if=%Aj. Hence in either case since ^' is other than zero, ^=0, B=0, C=0. Shew further that j is a perfect cube : that it is a factor of / : that i is a factor of j : that p is a, perfect fourth power, having J for a factor : and that jt? is a factor of H. Ex. (iv). If all the invariants of a quintic vanish shew that a must vanish and that j must be a perfect cube and a factor of /. Ex. (v). If r, = {xi/), i = a,^-^a„ f=a^^, then (i) n = % f-fi2/) = e + WJf-r,. (ii) n=Z, (iii) «=4, (iv) 71 = 6, {aehsch). [The student, who wishes for information concerning special quintics — when some of the invariants vanish — is referred to Clebsch, Bindren Formen, ch. VIII., and to Elliott, Algebra of Qualities, ch. xili.] CHAPTER VIII. SIMULTANEOUS SYSTEMS. 135. It was proved in Chap. vi. § 103, that if Si, 82 be any two finite and complete systems of forms, then the system S formed by taking transvectants of powers and products of powers of forms of Si with powers and products of powers of forms of ^S^a is both finite and complete. If S-^, S2 be the complete systems for any two binary forms fx^fi', then S is the complete system of concomitants for the forms yi, ^a taken simultaneously; for S is complete and contains both f^ and f^. Hence the complete irreducible system of concomitants of a pair of binary forms is finite. 136. To make the matter clearer, let us briefly recapitulate the argument. (i) Any concomitant of the simultaneous system can be expressed as a sum of symbolical products ; the factors in which are all of the following types (a6), (a^), (aa), a^, a^: where letters of the Roman alphabet refer to the quantic /j, and letters of the Greek alphabet to the quantic^. (ii) Any concomitant of the simultaneous system can be expressed linearly in terms of transvectants of products of forms belonging to the complete system for/j, with products of forms belonging to the complete system for f^. For any symbolical product in which the letters are partly Roman and partly Greek is a term of a transvectant ( U, Vy, where U is &, product containing only Roman letters and V a product 135-137] SIMULTANEOUS SYSTEMS 159 containing only Greek letters. But by § 51 any term of the transvectant {U, Vy is equal to {U, vy-¥t\{U, vy, where U, V are obtained by convolution from U, V respectively ; X. is numerical and p is less than p. Now U, U axe covariants of/j and hence maybe expressed as a sum of products of the irreducible forms of /i ; similarly V, Fmay be expressed as a sum of products of the irreducible forms of^. Hence the theorem is true for any symbolical product, the letters of which refer some to/i and some to/g : and therefore it is true for any concomitant of the simultaneous system. (iii) The system of transvectants {U, Vy, where ?7 is a product of concomitants of/i and V a product of the concomitants of /a, is both finite and complete. This was proved in § 103. 137. The complete irreducible system of concomitants of a finite number of quantics is finite. The proof of this theorem is inductive. Let us suppose that it has been proved that the complete system of concomitants of any n quantics is finite. Consider a set of ?i + 1 quantics, y 1 > y 2 > • • • /n+i • The n quantics ftf, ••• fn possess, by hypothesis, a finite system of concomitants which may be called 8-^. The single form /„^.i also possesses a finite system of concomitants, which may be called ^2- The complete system, 8, of concomitants of the n + \ quantics is obtained by combining 8-i^ with S^. And since the systems ^1 and 82 are both finite and complete, it follows that the complete system 8 is finite. Hence if the complete system of concomitants of any n quantics is finite then that for any w + 1 quantics is also finite. But the complete system for any one or any two quantics is finite. Hence the complete system of concomitants for any finite number of quantics is finite. We proceed to find the complete systems, in a few of the simpler cases. 160 THE ALGEBRA OF INVARIANTS [CH. VIII 138. Linear form and quadratic. The complete system of concomitants of a quadratic / consists simply of the quadratic itself and the invariant {f,/y\ Thus the system S^ is The system S^ — of the linear form I — is simply I. The combined system 8 is obtained by taking all possible transvectants, But unless /3 = 0, this is equal to JD^if^Jyy, and is certainly reducible. Again (/«, ^)« = (/^ Py . ly-\ which is reducible unless y = B. Further {f°; l^Y contains a term i/jy-if-M'-'f-', and is reducible if S > 2. The system 8 then consists of / A, I, (/ 0, (/ 0^. 138 A. Linear form and any finite system. Let the finite system referred to be denoted by Si. The system S^ consists simply of the linear form /. Let the system ^i consist of the forms C^, 0^, ... C^ which are of orders Si, s^, ... Sa- Then' we have to consider all possible transvectants If /9 > 7 this transvectant contains a factor I, and is therefore reducible. It may then be supposed that yS = 7- Let us suppose that a^ df ; then if y>Sr the transvectant contains the terms 138-139 a] simultaneous systems 161 and is reducible. If 7 :f> 5^ it contains the term (6V , ly)y . Ci-' Ca"* . . . C/r-^ . . . Ox^ and is therefore reducible. Hence the system S contains the forms Ci, C^, ... Ca ; I; (Or, 1^)^, 7 = 1, 2, ... Sr, r=l,2,...\; and these forms only. Ex. Prove that the complete system for a linear form I, and a given finite system of forms, consists of the linear form, the given system, and the forms obtained by operating with powers of (l ^-l ^\ on the members of the given system. 139. Two quadratics. Let /j, f^ be the two quadratics. We have to combine the two systems *Sfi, 8^, where 8^ consists of and (^2 consists of Since D^ and Dg are invariants, they give rise to no new forms. Hence we have only to consider transvectants The only irreducible transvectants which can be obtained are 'J\i = \f\ > J 2), and D,^ = {f„f^y. The required system is then Jli J2} 'J\2j ^1) -^2> -^12- 139 A. Any number of quadratics. Consider first three quadratics fi, f^, f^. To obtain their simultaneous system we combine the system 8-^ for /i, f^ with the system 8^ for f^. Leaving invariants out of account we must consider all transvectants (/l"//A.2,//)^ G. & Y. 11 162 THE ALGEBRA OF INVARIANTS [CH. VIII Since all the forms are quadratics the only irreducible trans- vectants obtainable are \f\> Js)) (/a. Jsh Wi2> Js) i \f\i fsfi if a, Jsf, («^i2> fzf- Of these {Jn,fz) is reducible, for it is the Jacobian of a Jacobian and another form. The rest are ♦'ISj «'23> -^13> -^23> and another invariant which may be called -£^123- The complete system for the three quadratics is then seen to be J\'J^>j3y *^12> «'I3> •■'23> "li -^2> -^S> -^12> -^23> -^31 > -^123- There is only one form of a new kind, and this is an invariant. Hence in forming the system for four quadratics, we shall not meet with any new kind of concomitant. And in fact it is easy to see that every irreducible concomitant in the system for any number of quadratics belongs to one or other of the types /, /, D, E. 139 B. It is easy to obtain the syzygies between the forms of the last paragraph. First J^^ is a Jacobian, and therefore, § 78 2/^,=-A/2^-A/l^^-2A2/l/2 (1). It MFill be convenient to use the notation ji^^ ^x > /2 — ^a; > y 3 — Ca;', .... Then as in § 77 we obtain 2 /i2 J34 = 2 {ah) aj)x . {cd) c^dx {acf (ad)^ a^ (bcY (bdy K' cj dx' = - A3/2/4 - DuAA + A4/2/3 + D^fJ, (2). By replacing /< by/i, a syzygy for 2Jj2Jgi is obtained. 139 A-139 b] Again hence, as in § 77, SIMULTANEOUS SYSTEMS ^123 = — (ab) (be) (ca), 163 2-£^123 • -C^456 — (ady (aey (a/y {bdy (bey (b/y (cdy (cey (c/y Du A5 Ae D^ D^ n^ D» A« Dss (3). From this may be obtained syzygies for 2 E\3s , 2 J&123 . ^124 . 2 £'i23 . Ey Similarly, the syzygy 123"46 — A. Aa A D^ D^ A Du D^ fs (4) may be obtained, and other particular cases may be deduced. Again h^ 6162 bi 6a;' c?i=^ d^d^ di di = 0, for the last column of this determinant is a sum of multiples of the first three columns. But it has been shewn, § 77, that -^123 = — {oh) (be) (ca) = (h' Oitta ai V 6162 bi C^' CiC^ (^ Hence fiEn^— f^Ei^+ f^E^^— fiE^^ = (5). If in the above determinant the elements of the last column are replaced by (aey, (bey, (cey, (dey respectively, another syzygy is obtained, A5-^234 — D^E^ + D^Eyn - D^Eva =0 (6). 11—2 164 THE ALGEBRA OF INVARIANTS [CH. VIII and In § 77, it was proved that {aef wy {^gy (ahy {hey (cey {b/y (c/y ((^gy {bhy (chy = (dey (d/y (dgy (dhy Similarly we obtain the syzygies A5 Ac Az / D^ D^ n^ D. /a = D^ D^ A7 A Du Aa Aa A D^ D^ A5 D^ B^ A A = A /» /« (7) (8), (9). Every kind of syzygy which occurs in the irreducible system of concomitants for any number of quadratics has now been obtained. Ex. (i). Shew that the last three syzygies just written dowu are not independent of those which come before, but may be obtained from them on multiplying by forms of the types E and J. Ex. (ii). Obtain the syzygies (1), (2), (5) by means of Stroh's method. Ex. (iii). Obtain (4) from (3), and (6) from (5) by transvection. 140. Quadratic and cubic. Let 4> be the quadratic and / the cubic. Then we have to combine the systems of forms Si and S^; where ^1 contains and S2 contains / (Afy = H, (f,H) = T, (H,Hy = A. All transvectants must be considered. Any transvectant for which either /S or 97 is other than zero is obviously reducible, it may then be supposed that ^ = 0,77 = 0. 139B-141] SIMULTANEOUS SYSTEMS 165 Again both <f> and H are quadratics, hence if B is not zero and f > 2 the transvectant contains the reducible term We have then only to discuss the transvectants (</), u), (0, Hf, (<^%/>ro^. Of the transvectants all contain reducible terms except {<\>,f\ (<!>>/)'> (<^^/n («^^/)''• Now since, by § 91, those transvectants for which e > 1 are all reducible. Hence of the transvectants all are reducible except for (<^, T) is reducible by § 77. The only other irreducible transvectant is readily seen to be ir, fTf. The simultaneous system for the quadratic and the cubic / then consists of: five invariants D, A, {<^,H)\ (4>^f% {4>\fTf\ four linear covariants three quadratic covariants <f>, H, (</), H) ; three cubic covariants /. T, (<!>,/). 141. Quadratic and any system of forms. Let the system of forms referred to be denoted by Si, the system S^ for the quadratic / consists of 166 THE ALGEBRA OF INVARIANTS [CH. VIII The invariants of both systems may be left out of account as they produce no new forms. All possible transvectants must be discussed, where P is a product of the forms Giy C2, ... Cx of the system Si. If p <2i — 1, U contains the term and is therefore reducible. Since p cannot be greater than 2r we may confine ourselves to the cases p = 2r, 2r - 1. If P be a product of two factors one of which is of even order, then U is reducible. For let P = P1P2, where Pj is of order 2t, then (p,p,, pr contains the term (p„/r.(P2./'-')^^ and is in consequence reducible. Also if P is of order > p, and is a product of two factors, JJ is reducible. By what we have just proved, if one of the factors of P is of even order JJ is reducible ; let then, P — P^P^ where P^ is of order 2^i + 1 and P3 of order 2^2 + 1, then {PiP,, fry contains the term (p„/^r..(p„/'-'.)'-^ since /a < 2^ + 2<2 + 2, and therefore p — 2ti< 2t<i + 2. Thus JJ must always be reducible except when P consists of a single term C; or when P consists of a product of two terms Ci, Cj each of which is of odd order, their total order being p = 2r, so that U={CiCj,frr. 141-142] SIMULTANEOUS SYSTEMS 167 Thus the irreducible forms belong to three classes : (i) {c, f^r-\ • (ii) (c, rr, (iii) (c,c,-,/r, where Gi is of order 2^+1, and Gj of order 2r — 2^ — 1 ; this latter class furnishes invariants only. It has not been proved that all transvectants belonging to these three classes are irreducible ; on the contrary we proceed to examine a case in which certain of the transvectants thus retained are reducible. 142. Let C he a, Jacobian = ((7;, Cm). (i) Let Ci = <f>^'-, C^ = t^^ /= a,^ = 6/ = . . . , G = ((jyyjr) <l>,'-'irJ^^-\ Then the form (G, f)^-' is reducible. For if 2r>2cr + 2T— 1, this transvectant vanishes; and if 2r <2a + 2t — 1 it contains the term {yjrc <" y {yjrc <=" y ... {yjrc <''-^-'' y 0a;2<r-2A-2 .,|^^2T+2\-!!r_ But {<f>y^) (<f>a) a^f^ = ^ [- {a^y<^,' + (<^^)W + {4>ciy>]r,'\ And hence the term written down is where T is a term of ((C^, G^y,f'-T'~^. (ii) If c^=<^,^ c^=^/r,^+^ then the transvectant (G, fryr-i vanishes if 2r> 2a- + 2t, and if 2r < 2o- + 2t it contains the reducible term («^.^)(<^a)a^t^(</,6<")^..(#<"')'^ {fc^^^y... (slrC^'-^-^^y <j)^^-'-^-^ ^^2T+2A-2r+i^ We are left with the case 2r = 2o- + 2t. 168 THE ALGEBRA OF INVARIANTS [CH. VIII (iii) If Gi = 4>^^^\ Cm = ylrJ^^\ then the transvectant (c, rr-' vanishes if 2r >2a- + 2t + 1, and contains a reducible term if 2r < 2o- + 2t ; but not if 2r = 2o- + 2t. Hence : " The transvectant (G, f"")"^-^ is reducible, if C is a Jacobian, except when one at least of the forms of which G is composed is of odd order, and the order of G itself is equal to 2r or 2r - 1." 143. Quadratic and Quartic. The simultaneous system of irreducible concomitants when the ground-forms are a quadratic and a quartic may now be written down. The complete system for the quartic <f) is known to be cf>, H = (<f>,<f>y, T = {<f>,H), i = (<f>,<l>)\ j^{H,4>f. Since there are no forms here of odd order, there can arise DO irreducible concomitants belonging to the third of the three classes mentioned above. The simultaneous forms are (0,/), (<^,/n {<i>^f% (<!>>/% (H,f), {H,f)\ (H,f% (E,f% {T,fy, {T,f% {T,f^f. It follows from the theorem of § 142 that the forms {T, f), (T, f% {T, py are reducible, T being a Jacobian. To complete the simiiltaneous system we must take into account the forms which belong to the quartic and quadratic separately; thus in all we have 18 concomitants. Ex. Prove that all the forms of the complete system for the two co- variants J and i of a binary quintic /, considered as separate quantics, are irreducible when considered as concomitants of the quintic ; with the single exception of one invariant of degree 18 in the coefficients of /. CHAPTEE IX. HILBERT'S THEOREM. Hubert's Proof of Gordans Theorem. 144. We shall now give another proof of Gordan's theorem that the irreducible system of invariants and covariants of any number of binary forms is finite. The method, which is due to Hilbert*, is of more general application than that of Gordan, inasmuch as with slight and non-essential modifications it applies to forms with any number of variables ; on the other hand, unlike Gordan's process it gives practically no information as to the actual determination of the finite system whose existence it establishes, in other words it proves that the problem always has a solution, while the other method, although only proving this for binary forms, gives much information as to the nature of the solution. In the exposition of Hilbert's proof we shall confine ourselves to binary forms, and to save trouble we shall deal with pure invariants only ; inasmuch as the complete system of invariants and covariants of any number of forms is really equivalent to the system of invariants of the set of forms obtained by adjoining an arbitrary linear form to the original set, the proof for invariants is sufficient for the most general case. Cf, § 139. 145. The proof may conveniently be divided into two parts of the following purport. I. Proof of the fact that any invariant I of the system may be written in the form 1 = -Aj/ 1 + A<il^ + . . . + -A-n^ny * Math. Ann. xxxvi. Story, Math. Ann. xlii. 170 THE ALGEBRA OF INVARIANTS [CH. IX where /j , I^... In o.re a finite number of fixed invariants of the system, and the A's while not necessarily invariants are integral functions of the coefiicients. II. The application to both sides of the equation just given of a differential operator which leaves an invariant unaltered except for a numerical multiplier, and changes a term, Arlf into one of the form Jrlr, where Jr is an invariant. As a result of I. and II. any invariant may be obtained in the form -t i«/i + Ii^Ji + . . . + J-n^n- Then by applying the same argument to the J's and so on it follows at once that the 7's form the complete system. 146. As a matter of fact the result I. is a particular case of a much more general proposition which we shall first enunciate, then illustrate, and finally prove. Theorem. If a homogeneous function of any number of variables be formed according to any definite laws, then, although there may be an infinite number of functions F satisfying the conditions laid down, nevertheless a finite number F^, F^, ... F^ can always be chosen so that any other F can be written in the form F=A,F, + A^F^-\-.,.-\-A,F,, where the A's are homogeneous integral functions of the variables but do not necessarily satisfy the conditions for the F's. Suppose for example that we have three variables x, y, z which we take to represent coordinates and that\^=0 represents a curve through the point 2/ = 0, z = ^ (this being the law according to which F is formed), then F may be written in the form 2/^ + ^Q. as follows at once since the highest power of x must be wanting in the equation ; 2/ = 0, 2^ = being two curves of the system, this is the application of the theorem to this case. As another example, if the curve pass through all the vertices of the fundamental triangle its equation may be written yzP + zxQ, + xyK — 0, 145-146] hilbert's theorem 171 where P, Q, R are integral functions of the coordinates, and here yz = {) etc. are curves of the system. Again, we have the famous theorem that the equation of any curve through all the points common to ^ = and ■y^ = may be written In each of these cases it will be noted that the system of forms F^, Fo.,...F^ is determined; in the general case it is not actually determined, the essential point being that it is finite and that the -4's are integral functions. To establish the theorem in its general form we first remark that it is manifestly true when there is only one variable x, because in this case each F consists of a power of x and therefore all the ^'s are divisible by that which is of lowest degree ; thus there is only one form in the system jF,, F.^, ... Fr. We now assume that the theorem is true when there are n — \ variables and deduce that it is true when there are n variables. Let a?i, x.^,...Xn be the variables and first suppose that the system contains a form H. of order r in which the coefficient of «;„*" does not vanish. Then we can divide any form in which x^ occurs to a power equal to or greater than r hy H without introducing coefficients fractional in the xs, and we can continue the process until the remainder contains no power of x^ higher than the (r-l)th. Hence we can write any form of the system thus F^HP + Mxn'-' + N, where P is the quotient, ilf is a function of x^, x^, ... x^-t,, and iV^ is a function of the variables but of degree r — 2 at the most in x^. Now the functions M are formed according to definite laws if the jP's are, because each M is deduced from the corresponding F by a definite process, and as they only contain n—\ variables the theorem is true by hypothesis for them. Accordingly we can choose a finite number of ilf's, say ilfj, M<i,...Mk, such that any other may be written in the form M = B,M, + B.,M, + ...+ BJI^, where the B's are integral functions of x^, x.2, ...Xn-i. 172 THE ALGEBRA OF INVARIANTS [CH. IX But since cCr:-^M=F-HP-N, Xrr'M,==F,-HP,-N„ etc., we have + B,{F,-HP,-N,), or F==H(P-B,P,-B,P,-...-B,P,) + B,F, + ... + B,F, + N-B,N,-,..-B,N,. Now the part of the right-hand side which does not contain one of the forms as a factor consists of B's and N*s and there- fore only contains Xn to the power r — 2 at most. Hence we may write F= HQ, + B,F, + B,F,+ ...+ 5, F, -[- if' ^»^/-^ + N^'\ and now il/w is a function of iCj, iCg, ... a;„_i formed according to definite laws and iV^''* is of order r — 3 at most in ic„. Thus we can write F as the sum of a finite number of terms each containing a form of the system for factor together with expressions of order 1 — 2 at most in the last variable. Then applying precisely the same argument to the Jlf *^*'s as we applied to the M's we see. that by adding a finite number of F's to we can reduce the order of the remaining portion in a?„ to r — 3. Proceeding in this way and adding only a finite number of F's at each step we can finally write F in the form where il/''"' only involves Xi,^^, ... Xn-i and in the nature of things is formed according to definite laws. . Hence applying the same process to the Jf' '"''s as we applied to the M's, we finally have F in the form AyF^-\-A^F^-it ...+AsFg, where the .^'s include H and the number s is finite. Consequently if the theorem be itrue for n-\ variables it is true for n, but it is true for one variable, therefore by induction it is true universally. 146-148] hilbert's theorem 173 We have now to remove the limitation imposed above, viz., that there exists a form of the system in which the coefficient of the highest power of ar„ is not zero. If there is no such form among the F's let Ft be one of the forms and apply to all a linear substitution iVr = ftriyi + 0,^2^2 + •.. + a„,y„ ; r = 1, 2, . . . w. Suppose that Ft (x) becomes Gt {y), then the coefficient of the highest power of y„ in Gt is Ft {am, ci^, • • ■ a^n). ^'"d therefore unless Ft is identically zero we can choose the linear substitution so that this coefficient is not zero*. Hence the theorem is true for the forms G in the variables y, and therefore by changing the variables back again from y to x we see that it is true for the F's. Q. E. D. 147. Returning now to the consideration of invariants it is clear that such an expression regarded as a homogeneous function of the coefficients of the forms is formed according to definite laws ; hence, if / be any invariant of the system, we have where 7i, /a, . . . /,iare n fixed invariants and the A's are homogeneous integral functions of the coefficients but not necessarily invariants. As to the functions A a simple remark may be added. All that is asserted in the general statement of the foregoing theorem is that they are homogeneous in all the coefficients, but an invariant is homogeneous in each set of coefficients involved taken separately, and consequently since / and Im are homogeneous in each set of coefficients, J.,„ is also homogeneous in each set. That this is so could of course be seen in the proof of the general theorem because at no point of the investigation is the homogeneity disturbed. 148. We now come to the second part of the proof, but before proceeding with it we must prove a necessary lemma on the properties of the operator D, so often used in the course of this work. If P be a function of ^i, ^2> Vi> V-2 which is homogeneous and of order X in |i, fg and homogeneous and of order yu, in rj^, rj^, then ^m (^j)np^ ^ CoD"-'" P + C,i)'»-'»+i n (P) + . . . + C,„i)"n"» (P), • We assume here that unless a form vanishes identically values of the variables can be found for which it is not zero. It is easy to give a formal proof of this theorem. Cf. Weber's Algebra, Vol. i. p. 457. 174 THE ALGEBRA OF INVARIANTS [CH. IX where J) = ^iV2 — ^2Vi> ^ ^^'^ ^ ^-re positive integers and the C's are either zero or constant. The result can be readily proved by induction, for we have „ dP ^ dP ^ d^P f J, ydP dP J. d'P \ and by Euler's theorem for homogeneous functions the right-hand side becomes (\ + fi + 2)P + nnp. Now in this result change P into D^^~^P so that X and fi are increased by n — 1, and we have O {D^P) = (X + yit + 2w) i)"-i P + DD. (i)"-i P). Hence Da (D"-'P) = {X + fi + 2n-2) D^'-^P + Dm (D^-^P), D^n (jy^-^P) = (\ + /I + 2w - 4) i)»-ip + ifD. (i)»-=^p), 2)«-2n (D^P) = (\ + /i, + 4) V'-^p + D»-in (DP), D"-^n (DP) = (\ + /I + 2) D«-i P + D«n (P). By adding these results together we obtain a {D'P) = {w (\ + /i) + n (w + 1)1 D"-iP + D»n (P), which establishes the result when m = 1 for all values of n. Assume that the result is true for any value of m so that 0,-^{jy^P) = CoD«-'»P + C'iD«-'«+ia (P) + . . . + C^D^n^ (P), then operating again with a we have r—m a"*+HD"P)= 2 c^a{D»-'«+'-a'-(P)}. But a(D"-™+'"P) = (w - m + r) (\ + /i + n - m + r + 1) D^-^'+'-^P + D"-*"+''a (P), 148-149] hilbert's theorem 175 and changing P into I1''(P) so that \, fi are each diminished by r, we deduce = {n-m+r){X+fM-r-m + n + 1) D«-'»+'-in'-(P) + X)n-m+r^r+i (p) when a^ is numerical. Thus we have r=l in other words, if the result be true for m it is true for m + 1, for the right-hand side is of the stipulated form. Hence by induction the theorem is true universally. > Ex. Prove that rX'^J i^+fi' + n-m + l)! {n-m + r)[ Cor. It clearly follows that if in the formal statement any exponent of D on the right-hand side be negative the corresponding coefficient is zero because only integral functions can appear in the process. 149. With the aid of the above lemma the proof of Gordau's theorem may be easily completed. For the sake of convenience we shall regard Xi, x^ as the variables in the fundamental binary forms, despite the fact that in the general theorem proved above they play the role that the coefficients do in the remaining portion of the investigation. Suppose the variables are changed by the linear transformation X^ = 52^^! "r Vii'^2 then an invariant I of the forms becomes (^vYL Further we have /= AJi + A0J2 + ...+ Anin and an invariant /,„ on the right becomes i^vY-'I.n. 176 THE ALGEBRA OF INVARIANTS [CH, IX We now write down the identity which is the transformation of I = A.^li + -"2-* 2 + ... + A.filn, i.e. the same identity for the transformed quantics; we suppose a coefficient Am written in symbolical letters entirely, so that it is the sum of a number of terms each of which contains only factors of the types a^ and a,, where a is a symbol belonging to one of the quantics. If after transformation Am become Bm we have 1 and the equation shews at once that Bm is of order /a — /*,„ in both ^ and 7). Now operate on both sides of this identity with _/ 8" 8" Y The left-hand side becomes a numerical multiple of /, viz. (fjb + l){fi \yl, and on the right-hand side we have n>^[{^r,y'^Bm]Im = Tm {Coi^vr-'-^Bm + C^i^vY-'^+'^B,,, + ... + G^i^y-a^Bm} by the lemma, since Im does not involve ^ or rj. But if fi — /i-m = V, then are all negative. Consequently t7o, (7|, ... Gy_i are all zero. Again J5,„ is of order /x, — fim = v in both ^ and tj, hence a-'+H^^), Xl-'+H^m), ... X>(5m) are all zero, and the effect of the operator on (^VY-Bm therefore reduces to a single term, namely G,£V{Bm). 149-150] hilbert's theorem 177 Now Bm is the sura of a number of terms each containing V factors of the type a^ and r factors of the type a,, hence by a fundamental theorem, is an invariant of the system. Therefore after operating with H** on both sides of the equation we are left with C I = lJmTm, where Jm= . ^.. Q,''{B„,) and is an invariant. Since Jm is an invariant we can express it also as the sum of a number of terms each containing an I^ as a factor, hence by continual reduction we can ultimately express / as a rational integral function of /j, I^,...Im, that is to say, these invariants constitute a complete system and, as we have seen, their number is finite ; Gordan's theorem is thus completely established. 150. Syzygies between the irreducible invariants. Examples of relations between the members of an irreducible system of invariants or covariants have already been given, and in fact a very large number were obtained for the quintic. It can be deduced from Hilbert's Lemma that the system of syzygies is finite, that is to say if >Si = be any syzygy we can find a finite number of syzygies ^, = 0, ^, = 0, ...^, = 0, such that 8= C^S^^- C^S^ + ... + G^S^, where Ci,...Cr are invariants. If there be such a relation, then of course all other syzygies are necessary consequences of S, = 0, S^=0,...Sr=0, and these constitute the finite system. The proof is very simple. Let /j, I.^, ... Im^e the members of the irreducible system of invariants, then S is a, function of Ii, I2, ... Im formed according to the law that it must vanish when for the /'s we substitute their actual values in terms of the coefficients. G. & Y. 12 178 THE ALGEBRA OF INVARIANTS [CH. IX Hence we have S ^ GiSi + t/jOa + ... + C^Sj., where Si = 0, etc. are a finite number of syzygies and the C's, being functions of the I's, are invariants. (Cf. § 127.) 151. Gordan's Proof of Hilbert's Lemma. Many versions have been given of the fundamental lemma of Hilbert on functions formed according to given laws, but the majority of them do not differ materially from the original proof due to Hilbert. Nevertheless Gordan has recently given a demonstration* which is so interesting and depends on such simple principles that we cannot refrain from giving an account of it here. We shall state it in the form of two theorems. Theorem I. If a simple prodtict of positive integral powers of n letters a;i*'a?2** ... «:«*", be formed in such a way that the exponents k^, k2, ...k^ satisfy certain prescribed conditions, then, although the number of products satisfying the conditions may be infinite, yet a finite number of them can be chosen so that every other is divisible by one at least of this finite number. To illustrate the scope of this theorem take the case of products of three letters and suppose the conditions are k, = 0{mod.S), Atg ^ "'s ^^ ' • The simple products satisfying the conditions are 1*2 '*'3 > "'2 "'3 > • • • /*«3/jri3/)f»2 /]/% a /ft 4 ni. 9 .*/j U/^ It's > «*'l •*'2 It's . • . > and it is evident that all such products are divisible by x^^a;^^ Again, suppose the sole condition is A'l A^2 ' "'3 ^ " J the products are /*» ft* ' /y«2/v»2/jf.,y». /jn 2fy« Of* tr* f ft* ft* " /y»3 •^Ij '*'3 ) "'I > **'3 > "'1'*'3 J •*'l •*'2> J'i««'2"'3, «t'2"'3 > tX/j , . • . and all the products are divisible by x^ or a;,. * Liouville's Journal, 1900. 150-151] hilbert's theorem 179 Other examples could be given, but the above will suffice to shew the nature of the theorem which we now proceed to prove. If 71 = 1 the truth of the theorem is evident because all the products are powers of a single letter and are therefore divisible by that having the least exponent. We shall now assume that the result is true for n — 1 letters and prove that it is true for n letters. Let a^i^'a^a"* ••• ^n**" be a definite product F satisfying the given conditions and let x-^'^x^^ ... a;/" be a typical product K of the system. If K be not divisible by P one of the k's must be less than the corresponding a. Suppose that k^ < a^, then, consistently with this, k^ must have one of the values 0, 1, 2, ...a^-1. Hence if K be not divisible by P one of a number «! + a-i + • • • + f*M = -^ contingencies arises, viz. either ki has one of the values 0, 1, 2, ...aj — 1, or k.2 has one of the values 0, 1, 2, ...a2 — 1, etc. Suppose that k^ = in, and that this is the pth of the possible cases ; then the remaining exponents k^jk^, ... Ar^i, A^^+i ...k^ satisfy definite conditions which are obtained by making k^ = m in the original conditions. Let Kp = x^^ x^'"^ . . . x^^ . . . x^'^ be a product of the system for which kj. = m and write Then K'p contains only n — 1 letters and the exponents satisfy definite conditions, and when these are satisfied the exponents of Kp satisfy the original conditions. Hence by hypothesis a finite number of products of the type K'p can be found such that every other such product is divisible by one at least of these. 12—2 180 THE ALGEBRA OF INVARIANTS [CH. IX Denote this finite system by Lx, L2, ... Lap 80 that Kp is divisible by one at least of the Z's. Thus Kp = x^^K'p is divisible by one at least of the products which all belong to the original system of products because every L belongs to the subsidiary system. Denote these latter products by then in the pth of the N possible contingencies K is divisible by one of the products M '1' M <■•" M '"''^ ■'■"■ p y •'" p J • • • •'•" p • Now one of these N contingencies certainly does arise when K is not divisible by P, and hence K must be divisible by one of the products or else by P. The exponents of the ilf' s all satisfy the prescribed conditions and they are finite in number, hence if the theorem be true for n — 1 letters it is true for n letters, but it is true for one letter and hence by induction it is true universally. 152. Theorem II. If a system of homogeneous forms be con- structed according to given laws, then a finite number of definite form^ of the system can be chosen such that every other form of the system is an aggregate of terms each of which involves one of the finite number of forms as a factor, and the coefiUcients are integral in the variables. (Hilbert's Lemma.) Suppose in fact that Xi,X2, ...x^ are the variables and that (f> is a typical form of the system. Now construct an auxiliary system of functions ri of the same variables according to the law that a function is an rj function when it can be written in the form 17 = XA<f), the A'a being integral functions of the variables which make the 151-152] hilbert's theorem 181 right-hand side homogeneous, but otherwise unrestricted except that the number of terms on the right-hand side must be finite. The class of functions r) is infinitely more comprehensive than the class <^, and it possesses the important property that a function of the form ^Brj which is homogeneous in the variables is also an 17 function. Now in examining the functions 7; we arrange the terms of one of them of order r in such a way that x^ comes first and, generally, a term comes before a term when the first of the quantities which does not vanish is positive. In such a case we say that the term S is simpler than the term T and T is more complex than S, so that any function rj is arranged with its terms in ascending order of complexity. Now the functions rj being formed according to fixed laws, their first terms satisfy given conditions relating to the exponents, and hence by Theorem T. a finite number of 77 's, say 'r}i,7)2, ... rj^, can be chosen such that the first term of any other rj is divisible by the first term of one of these. Take any function 77 of the auxiliary system, and suppose its first term is divisible by the first term of t/^,. and that P^ is the quotient. Then v — PiVmi is an rj function with a more complex first term than rj because ij and Pi'>7m, have the same first term; if we denote this new function by t] '^* we have Next, if the first term of 7/ 'i' be divisible by that of rj^^ we have », (1) _ P „ 4. «, (2) and the first term of t] <^' is more complex than that of 77 '^>. Continuing this process of reduction we find 7; ('■-1' = PrVmr + V ^^^ and so on. 182 THE ALGEBRA OF INVARIANTS [CH. IX Now the first terms of are in ascending order of complexity, and hence the time must come when there is no 77 function of the same order as rj with a more complex first term than rj ""> ; in that case we have Hence where the 77 's on the right-hand side are all members of the finite system 77i,?72,...77p. Now the 7} system includes all the <^'s, moreover each 7; contains only a finite number of ^'s and hence every ^ can be expressed in the form where <^i, (f>2,...<f>r are the <^'s contained in the expression for Vi} V2> •'• Vp ai^d the A's are integral functions of the variables. 153. Remark. If all the conditions satisfied by k^, k^,.., k^ in Theorem I. be linear homogeneous equations, then the theorem establishes the existence of a finite number of solutions by means of which any other solution can be reduced. The difference of two solutions being now a solution, it follows that by continual reduction we can express all solutions of the linear equations in terms of a finite number — this is the result otherwise proved in § 97. * The ij's on the right being members of a finite system are finite in number ; hence even though the number of steps in the reduction be infinite, there can only be a finite number of terms on the right-hand side. The same ^ may of course occur in more than one term, but in that case we should add all such terms together. CHAPTER X. THE GEOMETRICAL INTERPRETATION OF BINARY FORMS. 154. Given two points of reference A, B on any straight line, the position of any other point P may be determined by the AP value of the ratio ^-^ of the A P B "i> distances of P from A and B. A convention as regards sign is necessary to complete the definition ; it is convenient to regard the ratio as positive if P lie between A and B, otherwise as negative. When a binary form of order n is equated to zero, the ratio — may have any one of n values. These determine n points on the AP straight line AB, such that the ratio ^^ for each of these points is equal to one of the roots of the equation for — . The coordi- nates {x^, x^ then define the position of P on the straight line by means of the equation AP ^x, PB ~ x^' It is found advisable, as will be evident immediately, to define the position of the point P whose coordinates are (i^i, ^2) t>y means of the equation AP_.x, PB~^xl' where A, is a fixed numerical multiplier. A further convention will be useful, viz. the positive direction of measurement from A is towards B and that from B is towards A. 184 THE ALGEBRA OF INVARIANTS [CH. X To find the distance between {x^ , x^ and {yi , y^), in terms of the length I of AB. Denoting the two points by P and Q, we have AP^PB^ AB _ I \Xi x^ Xxi + x^ \Xi + X2 ' and similarly QB^ I 2/2 ^yi + y-z' Hence pq - {PB - QB) - ^_-^^___^^^__^ XI (yx) (^2/1 + ^2) (5^1 + 573) 155. Let us consider the effect of a change in the points of reference. Let the new points of reference A', B in terms of the original system of coordinates be (^1, ^i), (171, tj^); if the new multiplier be fi and {Xi, X^) be the new coordinates of P, then X, A'P X,~^PB" ^^^1 + ^2' (v^) Hence The change in coordinates is thus equivalent to the linear transformation Xi = pi (x^), where -=/'^l4f- 156. A linear transformation X2 = ^2X1 + 7)2X2, may be regarded geometrically from two different points of view : (i) As changing the points of reference and the constant multiplier, but leaving the other points on the straight line in their original position. (ii) If the points of reference are regarded as fixed, the 154-156] GEOMETRICAL INTERPRETATION OF BINARY FORMS 185 transformation alters the positions of all the points defined by the algebraic forms under discussion. Consider the first of these points of view. When x^ = 0, the point P(oei, x^ coincides with one of the points of reference A. Similarly, if a^j = 0, P coincides with B. Hence to find the new points of reference in the original system of coordinates, it is only necessary to write X-^ — 0, and X2 = 0. We obtain them at once as (^1, t]^ and (fi, fg)- The distances of P from these new points of reference A\ B' are \W. ^?^^- .=u (^'^>^^ {\x^ + x^ {Xtji + 7/2) (A^i + ^2) (^^1 + %) ' and XI ^ ^.-— .. = X/ ^^'^^^^ (}U;i + x^) (X|i + I2) (\^i + x^) (Xf 1 + ^2) ' The ratio of these distances is A'P \^x + ^2 ^1 PB Xvi + V^'^ Hence the new multiplier is -^ — —; and the coordinates {Xi , X2) define the same point as that defined by the coordinates (xi, X2). It should be observed that the sign of the expression — ^^ -^ . ^ is positive if X lie between A' and B', otherwise it is negative. Ex. (i). Shew that by properly choosing quantities oj, ag the distance between the points (^1, x^), i^y, y^ maybe written ~^ . And that in this case the constant multiplier after transformation becomes ,— ^ . Ans. a^=-^ -, a^^sj J. Ex. (ii). The point (aj, a^ of the last example is the point at infinity on the range. When an invariant of a binary quantic is zero, there exists some relation between its roots which is unaffected by any linear transformation. Hence when the binary quantic is regarded as the analytical expression of n points on a range, the vanishing 186 THE ALGEBRA OF INVARIANTS [CH. X of an invariant is the condition that there may be some definite geometrical relation between the points, independent of the points of reference and of the constant multiplier. For example if two of the points coincide an invariant — the discriminant — vanishes. Again, as will be shewn later, if four points on a straight line which form a harmonic range are represented analytically by a quartic, then the invariant j of that quartic is zero. 157. In the second point of view stated in the last paragraph, the points of reference and multiplier are regarded as fixed, the point P takes a new position P' given by the coordinates Let the points Q, R, S, viz. (y^, y^, (z^, z^), (Wj, w^) become Q', R\ S\ Then PQ = xr, ^^ — , and (ya^) = (^v)(YX), Hence PQ^BS _{y^wz) PS.EQ~{ww)(yz) _ (rX)iWZ )_ P'Q'.R'S ' ~(WX)iYZ)~' P'S'.KQ'' The expression p^—pTs is called the cross (or anharmonic) ratio of the four points P, Q, R, S ; it is usually denoted by {PQRS] * The result just proved may be written {PQRS} = {P'Q'R'Sl * By rearranging the letters P, Q, R, S we obtain 24 such cross-ratios. It is easy to see, however, that only six of these are different. Then if X, fi, v are written . for the three products PQ . RS, PR . SQ, PS . QR, the six different cross-ratios are \ /l V fl u \ fl P \ \ fl V Since the four points are collinear, PQ.RS + PR.SQ + PS.QR = or X-f-^ + i' = 0; by means of this relation all six cross-ratios of four points may be expressed in terms of one of them. This mode of presenting the subject is due to Mr R. R. Webb. 156-157] GEOMETRICAL INTERPRETATION OF BINARY FORMS 187 That is, the cross-ratio of four points on the range is unaltered by any linear transformation. Hence the transformed range is homo- graphic with the original range. Further any range homographic with the original range may be obtained from it by a linear transformation. To prove this, it is only necessary to prove that the coefficients of transformation may be so chosen that three non-coincident points P, Q, R of the original range are changed to any three non-coincident points P', Q', R chosen at random on the straight line. For when P', Q, R' are known, the point S' of the transformed range corresponding to S is given by {P'Q'R'S'] = {PQRS]*. X Y Z Let us suppose then that the values of the ratios -^ ,^ , -^ are given. Then x. and or Similarly and ^2 7- . -' - ^1 ^ + '?2-^ - '/i = 0. ■"2 ■*2 ^2 ^2 We have three equations to determine the ratios of the coe ffi cients f 1 , |^2 , "^i , Vi • These ratios are thus determined uniquely. Hence a range of points on a straight line may be transformed into any other range homographic with itself by a linear transfor- mation. If an invariant of a binary quantic representing a range of n points is zero, these points must possess some special property, * This must give a unique position for S' since it is equivalent to a linear relation between its coordinates. ^1 + ..) X^, x^=\P., X, 'X2 + '72 X, ^ + '/! SC^ + »72 1. X, 'x. f4:+ •'72 % = = 0. ^2 2/2 4:+ « 2/1 y2 '71 = = 188 THE ALGEBRA OF INVARIANTS [CH. X which is also a property of all ranges homographic with the original range. Such a property is said to be projective, and thus the vanishing of an invariant must be the condition for the existence of some projective property of the points which the quantic represents. Conversely, if a system of n points on a straight line possesses some projective property, there will exist a corresponding analytical relation between the coefficients of the quantic repre- sented by these points, which is unaltered by any linear transformation. It does not necessarily follow that the condition is represented by the vanishing of an invariant; it soijietimes happens that a projective property necessitates the vanishing of all the coefficients of a covariant. Again a covariant of a quantic will define a certain number of points on the straight line. These points are related to the original points of the range in the same way as their homo- logues are related to the homologues of the original points on a homographic range. It is usual to denote this by saying that the points are projectively related to the points of the original range. 158. A binary form is homogeneous in two variables, we are then — in such forms — only concerned with the ratio of the variables. Let /{x^, x^) be any binary form of order n, then the equation f{x„x.,) = {) defines n values of the ratio — . Hence in any geometrical figure x^ in which the geometric element is completely defined in position by a single parameter, the iorici f {x^, x^) may be considered as defining n of these elements. For example a point which lies on a unicursal curve is such an element, li x, y, z are its Cartesian coordinates, it is well known that we may express x, y, z bs, rational algebraic functions of a single parameter. Again the tangent to a fixed unicursal curve may be taken to be the element. Or else the element might be the osculating plane of a twisted unicursal curve. Now the two simplest figures of the kind, are a range of points on a fixed straight line, and a pencil of straight lines 157-159] GEOMETRICAL INTERPRETATION OF BINARY FORMS 189 passing through a fixed point and lying in a fixed plane. We may deduce the properties of the latter from the former. For if any straight line be drawn to cut the pencil there is a one-to-one correspondence between the points on the range thus formed and the rays of the pencil. In fact any ray of the pencil may be defined by the coordinates (x^, x^) of the point in which it intersects the straight line. With this definition it appears that everything that has been said for the range applies equally well to the pencil. 159. A binary form may be expressed as a product of n linear factors. A covariant of the binary form is necessarily a covariant of the system of linear forms of which it is a product. Thus let aa;" = (a;a;W)(^^"')---(^«'"') (I). X '^* X "^' where -^ , ^r, , ... are the roots of the equation aJ^ = 0. Any invariant may be written in the form since it is an invariant of the linear forms (a;^*"), (ara;''") .... The coefficients of the quantic are given in terms of the roots by equating the different powers of x in (I). Two things are at once apparent. (i) The coefficients are symmetric functions of the quantities a;W, a?<=" .... (ii) The coefficients are functions homogeneous and linear in each of the n sets of variables a^i''', x^'^^ ; a;i<^', x./^^;... a?!*"', a^g*"'. It follows that any function of the coefficients must, when ex- pressed in terms of the quantities x^^\ a;'^' ..., be symmetrical in them. And further such a function must be homogeneous and of the same order in each of the sets of variables Xi^^\ x^^^^;... ^ («) ^ (n) Hence in the expression for an invariant (II) it is necessary that where ^ is a quantity which is the same for each terra of the sum representing the invariant. 190 THE ALGEBRA OF INVARIANTS [CH. X Let T be any one term of this sura, then let Then iSi2 + /3,3+...+^x„ = 0, Ai + /923+...+/32« = 0. We are going to prove that ^ is a function of anharmonic ratios of the roots. It will be assumed that when the number of quantities 3o^^\ a?® ... is less than n, then the term where 2/Si^ = 0, 1^^ = 0, r r ' may be expressed as a function of the anharmonic ratios. Now the ratio (^(1,^(2,^,4,^,0} =l^^)i£^. Hence ^ ^~^ ^ ''•(a;Wa;<=»>)'{ic<«ar<2>a;Wa;<*)}* On replacing (aj'^'a;'") wherever possible by the value just found becomes where P is a function of anharmonic ratios, and Q is of the form where S/Sar = 0, SySsr = 0. r r The theorem has been assumed true for Q, hence with this assumption it is true when there are n quantities a;<'>, a;*^', If there are only three quantities, then y3i2 + ^i3 = 0, /^Sl + ^32 = 0, and J3n = = ^^ = ^.^,. 159-161] GEOMETRICAL INTERPRETATION OF BINARY FORMS 191 Hence it is true when there are 4 quantities x, and therefore also when there are 5, and so on universally. Thus, any invariant of a binary form is the numerator of a rational function of the anharmonic ratios of the 7'oots. If the invariant contains only one term, there is an apparent exception. The invariant equated to zero then represents the condition for the equality of a pair of roots, it can only be the discriminant. 160. So far our remarks have been confined to the case of a single quantic ; a slight alteration in the wording of the previous paragraphs is all that is necessary to make them applicable to any system of binary forms. Each binary form of a system is geometrically represented by a set of points on a range, or of rays of a pencil. Points belonging to the same quantic must be regarded as indistinguishable from one another. Thus if we have a set of n points on a straight line, we may regard them as given by a single quantic of order n ; by two quantics of orders r and n — r respectively, or even by n separate linear forms. Now let two of these points coincide ; then, if the n points are regarded as a single quantic, the discriminant is zero ; but there is nothing to tell us which roots coincide. We may regard the n points as two quantics, in this case either the discriminant of one of the quantics or else the resultant of the two is zero. 161. We shall now discuss the geometrical representation of the invariants and covariants of the binary forms of lowest order. A quadratic has only one invariant, this vanishes when the points representing the quadratic are coincident; it is the dis- criminant. A pair of quadratics aj^, h^ have a simultaneous invariant {ah)\ Then if a;<i>, x^"^ are the roots of a^ and 3/^, y*^) those of h^, (abf = (a2/<i>) (ay^'^) = ^ [(a;("y<«) (a;'^'^*^)) + (a;<«y(«) (ar'^'y «)]• If {aby=(), it follows that (x^^)y(^))(xi^)yW)~ Hence the pair of points 3/*^', 1/'*' is harmonic with the pair 192 THE ALGEBRA OF INVARIANTS [CH. X The quadratics have a covariant, their Jacobian, ^ = {ab)axbx. Now (a^)^ = 0, (6^)'^ = 0, hence ^ represents the pair of points which is at the same time harmonic with a^^ and with bj?. In other words ^ represents the pair of double points of the involution defined by the two pairs a^^ bj". The discriminant of ^ is {^yy = i {{aaj (bbj - [{abm ; if this is zero, ^ is a perfect square ; the double points of the involution coincide. Hence, as may be verified either geometrically or algebraically, one of each of the pairs a^^, b^ coincides with the point represented by ^, and the other two points may be anywhere on the range. Thus a^, b^ have in this case a common point ; hence (^^')^ may be taken to be the resultant of the two quadratics. If there are more than two quadratics, there is only one more type of concomitant to be discussed ; viz. the invariant (a6) (6c) {ca). This is equal to — (^, c^y. If this is zero then the pair of points ^ is harmonic with the pair Cx- Hence (a6)(6c)(ca) = represents the condition that the three pairs of points a^, b^, c^ should be pairs in involution. To find the anharmonic ratio of the four points defined by O'x, bj^, we have (a;Wy(«) (a;®2/»') + (a;(i'y<«») (a;(2)y<i') = {ahf, = v^(a«7.V(667. Hence 2 (a;(i'y<«) (a;»»y W) = {aby + ^{aa'f.{bbj, 2 (a?»'y ('^O (a;<«y <«) = {ahf - ^/{aa')\{bb')\ and therefore ^ (aby + ^(aay.(bby {aby - ^{aaJAbUy' 161-163] GEOMETRICAL INTERPRETATION OF BINARY FORMS 193 . Denoting the anharnionic ratio by p, and squaring, this equation becomes D'^(p-iy=D.D"{p + iy-, or P'-^P j)'2_])j)- + ^ = ^' where i) = (aa')-, D' ={aby, D" = {bhy. The two values of p correspond to the two anharmonic ratios ja;(i)y(i)a;<2>2/<-'»} and {aj'^'y^^iai^/Wj. If ^ = 0, then the two quadratics are such that one is a multiple of the other. This is merely a particular case of the general property of the Jaeobian ; it is not necessary to do more than mention it here. 162. When a range possesses geometrical peculiarities which are unaffected by projection, there exist analytical relations of an invariant nature among the coefficients of the corresponding binary form ; but it must not be supposed that these relations can always be expressed in terms of the pure invariants of the form. If there is only one such relation which expresses the necessary and sufficient condition that the range may possess a certain projective property, then it will be found that A is an invariant, for it is unaltered by linear trans- formation. On the other hand, when the condition is expressed by a set of algebraical relations ^=0, 5 = 0,... A,B,... will not, in general, be invariants. Thus the condition that a binary form of order n may be a perfect nth. power is that all the coefficients of its Hessian vanish*. 163. The Cubic. Any three coUinear non-coincident points can be projected into any other three collinear non-coincident points ; it is not to be expected then, that the geometry of a binary cubic will be of much interest from a projective point of view. But in respect of the associated points furnished by covariants, the geometry of the binary cubic is highly interesting. * See later, Chap. xi. G. & Y. 13 194 THE ALGEBRA OF INVARIANTS [CH. X The single invariant A is its discriminant and A = is the necessary and sufficient condition that two of the three points represented by the cubic should coincide. If all three points coincide, then A, H and t are all identically zero, as may be easily verified ; but represents the necessary and sufficient condition*. 164. Let us consider the pencil where k and \ are new constants which determine the particular members of the pencil of cubics. Theny^ ^ represents three points, which are called a triad ; by varying k and \ a pencil of triads is obtained. The covariants of f^^^^ will be denoted by the symbols ■ffic,A> ^(c,\> ^K,\' These may be at once calculated; the following table, most of the results of which are proved in Chap, v., will be found useful for this purpose. Index Transvectant {f,.f) ifrS) (/,0 {H,H) {H,t) {t, t) 1 t -^m W 2 H A iAzr 3 A The fundamental forms, it will be remembered, are connected by one syzygy, t' = -^[H' + A/^}. To obtain H^x we have E^^x = {Kf-h\t, Kf+\ty = fc' (f,fr + 2«x (/ ty + x^ (t, ty = (/c" + i A\^) H. • Chap. XI. 163-165] GEOMETRICAL INTERPRETATION OF BINARY FORMS 195 Hence, if we use the notation H.,, = %H. In the same way, we obtain = © (/c« - ^ AX/) And lastly ^k.\= @'^- It is worth noticing that if we introduce the arguments /c, \ of (s) as suffixes, thus the syzygy may be written 165. Consider the relation if any pair of the three forms /, H, t have a common factor, then all three must have this factor. Let us suppose that such a factor exists, and let us change the variables so that the common factor is x.2. Then /is of the form (0, Oi, tta, tts^^i. «2)^ and the coefficient of x^ in H '\s —Oi; hence if x^ is a factor of H, we must have a^ = 0. This means that / has a double factor, and therefore A = 0. The syzygy then becomes whence it is easy to deduce that ^ is a perfect square and t a perfect cube. In fact if /= K% then H=A^,t = ^/^i:M^ = 0, where ^ = - I [(?<?)?• 13—2 196 THE ALGEBRA OF INVARIANTS [CH. X 166. It will now be supposed that f, H, t have no common factor. The syzygy may be written hence if ^, 77 are the factors of H, so determined that we may take 2p =\t-\- /y -^ ) ' and 2v^=(t-f^-^^y and therefore Hence also It is at once apparent that the only members of the pencil which possess double factors are ^^ and 7]\ Let Pi, P2, P3 be the three points determined by any one Tnember of the pencil, and A, B the two points determined by the Hessian (which are the same for every member of the pencil). Then if w be a cube root of unity, the points Pj, PzyP^ correspond to linear forms f — at), ^ — (oarj, ^ — o)-ar). The ranges A, A, P„ P,, B A, P,, P„ P„ B A, P3, P„ P„ B are projective. A set of points such as Pj, 2\, P, are said to be cyclically projective. 166-167] GEOMETRICAL INTERPRETATION OF BINARY FORMS 197 The simplest way to find the anharmonic ratio [APiP^Ps] is to transform the variables to ^, rj. This transformation may be regarded as merely changing the points of reference, and the constant multiplier. Then Hence the range formed by a triad and one of its Hessian points is equian harmonic. The six distinct cross ratios of such a range are each — to or — co^. 167. The Quartic. Just as in the case of the cubic we considered a pencil ot cubics instead of the single one, so here we shall find it convenient to consider the pencil instead of the single quartic f. Each member of the pencil will define four points, one of these points may be chosen at will on the range considered, but when this is done the ratio k :\ is fixed, and the remaining three points are uniquely determined. The calculation of the covariants of f^^x. presents no serious difficulty. For convenience a table of the transvectants of the quartic is appended ; most of the calculations were effected in Chap. V. Index Transvectant (/,/) if,H) iH,H) 1 (f,t) 1 {H,t) {t,t) 1 2 3 t i,iP-\m IfUf-iS) ff ¥f kJf-¥S yHf-i^im-^\i-^p kUf-iS) ii^f-\js: 4 i j ¥' The only transvectants which are not contained in this table are 198 THE ALGEBRA OF INVARIANTS [CH. X The syzygy between the forms is In connection with the system for/«_x the expression or more briefly H, will be found of great importance. We observe at once that the syzygy may be written Now _ / an an\ K.K = kH + 2k\j + \4 i" = - 3 (n, ny = — SHa, where fl is regarded as a binary cubic in k and \. Lastly r .an . / an ^dn\ , .^^ ani •^-^ = n"''9^^/H"a.-^axj + **^ax} 168. The invariant of the cubic CI is This, as we proceed to shew, may be taken to be the discriminant of the quartic. The discriminant is the condition that the equation /=0 may have a pair of equal roots. It is an invariant ; for if the range represented by / be projected, the pair of coincident points project into a pair of coincident points. Further it is well known to be of degree 167-168] GEOMETRICAL INTERPRETATION OF BINARY FORMS 199 2 (w — 1) for the quantic of order n ; hence it is of degree 6 for the quartic. It is then a linear function of i^ and j^. Let us suppose that f has a double linear factor ax, the remaining quadratic factor being p^^, and then find what relation exists between i and j for the quartic la the first place H={aa?.px\ a^J = \ {{aaypj" + {payaj" + 4 (aa) (pa) a^Px] ««' = i{S{aaypx' + S(payoix'-2{apyax']ax' (I). But {aafax' = (/, ax^f = i {apf . ax\ (par ax' = (/ px'Y = I [{oLpfp.' + 3 {p, pyax% Therefore H^-^{apff+l{p,pra\ Similarly i = (aa)- (apY =((oiayax\ px'y=h[(^pyf> i=(/. Hy=-i{^pr.i Therefore i^ - 6/ = 0. Hence An may be taken to be the discriminant of the quartic /*. From the above form for H, it is evident that H contains the factor ax twice over ; hence if / contains a repeated linear factor then every form of the pencil /c/4- ^H contains the same repeated factor. This leads us to expect that the discriminant oi Kf+\H is a multiple of the discriminant of /. This is so ; for from the syzygy for the cubic we obtain -^n«-2^n^ = An.n^ or (i,x^ - 6 j,, x') = ^' (*■' - Qj% It is easy to obtain in the same way the condition that / may have two pairs of repeated factors. For writing as before /= CLx' • Px^, we obtain t = {f, H) = iip,pY(f, a)a,^ 200 THE ALGEBRA OF INVARIANTS [CH. X Now if/ contains two pairs of repeated factors, (p, pf must vanish, for p,^ is a perfect square ; hence in this case t = Q. This is the necessary and sufficient condition, for if it is satisfied {t, ty=iQi^-f) = o, and we are at liberty to assume that f has one repeated factor ; then using the relation we see that either {p,py==o, in which case ^ is a perfect square ; or (/ «) = 0, in which case /= «.^ This furnishes another illustration of the remarks in § 162. Ex. (i). Shew that the necessary and sufficient condition that /may have a three times repeated factor is i=0, j=0. Ex. (ii). There are in general three difiFereut members of the pencil fg^x which are perfect squares. They are given by solving the equation £2=0. 169. As has been already pointed out, the syzygy for the quartic may be written If Wi, mj, mg be the roots of the cubic n = o, then fl,t, ,\ = ('c — niiX) (« — m^X) (k — wigX). Hence also 2t' {H + mj) (H + m,f) (H+mJ). If H and / have a common factor, by transformation this may be made x^. Then / is of the form (0, Ui, a^, a,, tti^Xi, x^y. 168-169] GEOMETRICAL INTERPRETATION OF BINARY FORMS 201 In order that a^a may be a factor of H, we must have Hence x<^ is a factor of/; and therefore An = 0. Exchiding this exceptional case, it is evident that no pair of the expressions H + Wi/, H + niof, H + m-if have a common factor (mj, m.^, m^ are distinct) for An is the discriminant of II. Hence the above relation shews that each of the expressions H + 'mf must be a perfect square — since f^ is a perfect square. Let H + m^f= — 2-v/r2 H-¥m,f^-2x\ where <^, -v/r, p^; are binary quadratics. Then t = 2<^^^x• As an example it may be verified that for the quartic H= 2ax^* + 2 ( 1 - 3a2) Xj^x^^ + 2ax.^, Q = K3-(3a2 + l)«X2-2a(l-a2)X3, and that the roots of Q = are a— 1, a+ 1, -2a ; which are identical with the values of m which make H+7nf a perfect square. Now (7/ + nil/, H + tiiof) = (nil — ^^2) t = (-2<^^ -2>|r2) = 4(</,, A|r)<^^; putting in the value of t we obtain 2(<^, -(/r) = (m, - ma) %. Similarly 2(^, X) = (m2-m3)«^, 2(x, <^) = (w3-mi)'»/r. 2 (rrh- - iri^) (wii ■ -TOg) 2 (ma- - ms) (to., ■ -TOi) 2 (W-j- - nh) (wig ■ -TOi) 202 THE ALGEBRA OF INVARIANTS [CH. X Now from the expression for the Jacobian of a Jacobian we obtain Hence by repeated use of this formula ={<t>xyf-(<f>n-x^ X = (4><f>) X-iM <!>' X = ii^^f X - i^xT '^' = ('^<f>T X - ("^xT </>' =ix-f)''t>-ix<t>)'f- Since cji, >/r, ^ have no common factors these equations give the following six relations {<}><l>y = _ |(mj - TO,) (tox - TO3), (fxy = 0, (i^y = - ^ (ma - TO3) (TO2 - mi), {x<l>y = 0, (XXY = - i (^^3 - m,) (m^ - TOa), ((fyyjry = 0. The remaining invariant of these three quadratics is ((}i^|r) (yfrx) (%</>) = - ((<^V^) <f>xirx, X^xf = - ^ (?ni - TOa) . ixxf = - i (Wli - Wla) (Wla - Wis) (wis - »Wi). 170. By means of the equations 5'^-TOl/=-2<^^ H + mJ=-2x' (II), the quartic /, or more generally Kf+ \H, may be separated into quadratic factors. 169-170] GEOMETKICAL INTERPRETATION OF BINARY FORMS 203 2 and Kf+\H= — {(k - \m,) %^ - (/c - Xm,) -f'] mj — Wg (V/e — Xm^ % + V/c — Xwis i/r) (V« — Xwg % — V/c — Xmg a^). The second transvectant of either of these quadratic factors with (f) is zero. Hence the two points determined by (f> are the harmonic conjugates of the two pairs of points represented by the above quadratic factors of Kf+ \H. Now we have only used the last two of equations (II) to find the quadratic factors of /c/'+ XH. Any pair of the three equations might be taken. The three results represent the three ways into which the quartic Kf+XH may be separated into quadratic factors. Then the three quadratic factors of t are the three pairs of points harmonically conjugate with respect to the four points Kf+XH, when divided into two pairs of points. Now (cf>r}ry = 0, {^Jrxy = 0, (x<^>^ = 0, hence the pair of points (f> is harmonically conjugate with respect to each of the pairs -v/r and x- If the points Kf+ Xll are divided into two pairs in any way, these pairs determine an involution, one of the quadratic factors of t represents the pair of double points of the involution. The other two quadratic factors of t represent pairs of points belonging to the involution. Now the points determined by t are independent of k and X, hence the pencil /c/"+ XH represents sets of four points such that when any set is separated into two pairs of points, these are pairs of one of three fixed involutions. The quartic / is arbitrary, it may represent any four points. Hence the pairs of double points of the three involutions deter- mined by four points on a line are harmonically conjugate two and two. 204 THE ALGEBRA OF INVARIANTS [CH. X 171. To determine the anharmonic ratio p of the four points /. We have obtained the quadratic factors ofy, one pair is X-'f' X + ^- The anharmonic ratio of the four points determined by a pair of quadratics has been obtained in § 161, as a root of the equation In our case = - ^ (mg - mi) (mg - m^ - ^ {m^ - w^ (m^ - m^) = -^(nis- m,y = D". D' = -^ (ms — mi) (mg - m^) + \ {m^ — m^) (m^ — m^). Hence . _ K- ^0^ + (^.- ^^O^ ^ ^ ^ ^ (mg - mj (ma — mj As there are six different values for the anharmonic ratio of four points, a sextic for p is to be expected. This will be obtained by multiplying together the three equations similar to the above. It will be more convenient to write these equations in the form (mg - mO (7^2 — mi) '^ Now mi, ma, mg are the roots of the cubic a;3 - ^ /«;X2 _ I X3 = 0. The discriminant of this is (mi - m^"^ {irru, - m^y (mg - mi)'^ = \ (i^ - 6f) = A, the exact expression is most quickly obtained by using the equation of the squared differences of the cubic. The equation whose roots are mi — m^, m^ — mg, mg — m.i is obtained thus S (nil — ^2) = 0, 2 (mg — mi) (mi — m^) = — '^m^ + Smamg = SSmamg = — 32, (mj — ma) (mg — mg) (wig — mi) = VA. 171] GEOMETRICAL INTERPRETATION OF BINARY FORMS The equation is ]f — '^ny~ "^ ^ — ^■ The equation whose roots are (7/ii — miY, ... is 205 or or But Hence (1-6 Therefore (z - VA)' = 27 ^ ^. " o (m., - W3V = - V A " P aM- p'-^P + l _iY = _ 27 ^' P'-^P + ^ ^,\ 8 1 + P J op p^-2p + lV*_p2-2p+2 27 p * 4 I 24(p2-p + l)=' j2 4(p^-p + l)^-27pnp-l)^ {p + inp-2y{2p-iy * When the quartic is not treated by the symbolical method it is usual to define the invariants as follows : — the quartic itself is f=(a, b, c, d, e\xi, Xj)'*, l = ae-Ud-^ 3c\ a b c bed c d e The invariants i, j in the text above differ from these by numerical factors only. Thus i=:{aby=2I, i = (aJ*)2(6c)2(ca)2 = 6/. In connection with the calculation of the values of invariants given symbolically in terms of the actual coefficients the reader may find it interesting to discover the fallacy in the following : — j=(6c)-(ca)2(ai>)2 = {{bc)(ca)(ab)}^ a,^ 61^ Ci2 0,2' 1)2 c^ aj* + bj* + Cj* = 27 J. ttjOg ijftg *'l<'-> 206 THE ALGEBRA OF INVARIANTS [CH. X Thus the anharmonic ratio is expressed by means of a sextic equation in terms of the absolute invariant ^ . We see from this equation at a glance that if i = 0, the points represented by the quartic form an equianharmonic range, for then, p'-p + 1=0. Similarly if j = 0, the four points form a harmonic range. Again, if two of the points of the range are coincident, one value of p is unity : hence as it should be. 172. The anharmonic ratio for the four points k/ + \H may be obtained at once by writing i^^. for i, and J^^^ for^'. To determine those values of « : X, for which the four points have any definite anharmonic ratio p; let a-21 C^-p-^P^y Then i\>.-aj\^ = 0, or 3Ha^ + atQ^=0, this is a sextic for k : X. Now ^n' = -2<n'-n2An. Hence {a-6)ta^ = n^Aa, «"=±"\/^ or .„-.-„, g The sextic thus reduces to two cubics. If a = 6, it is easy to see that p = 0,l, oo , hence two points must coincide. In this case H = 0, and - has one of the three A, values Wi, Wa, m,. 171-173] GEOMETRICAL INTERPRETATION OF BINARY FORMS 207 Hence the three members of the pencil for which a = 6 are H + m-^f= ^^ shewing that if one pair of points coincide, the other pair must also coincide. If a = 00, the four points form a harmonic range, and ^n = 0. There are three members of the pencil for which the range is harmonic. If a = 0, then Hq, = ; hence there are only two members of the pencil which form equianharmonic ranges. In all other cases, there are six members of the pencil having a definite anharmonic ratio. Ex. If IJx is the Hessian of the cubic a/, prove that the quartic a^Hg. is equianharmonic. 173. Case •when An = 0. The discussion was limited in § 169 to the case when An is other than zero. Now An is the discrimi- nant of the quartic /, and hence when An vanishes, two of the roots of /= are the same. We may, as in § 168, write where aj' is the square of a linear form ax, and p^^ is a quad- ratic. Then as before The invariant An is also the discriminant of the cubic fl = /r' - ^ «V - ^ V, hence, in the present case, fit has a repeated root. Let this be ma, and let the other root be nii. Then 2in2 4- mi = 0, Wg^ + 2mi?7i2 = — « > m^^mi = ~ . Hence »W2 = - 7 = i (apf, mi = / = - ^ (apY, 208 THE ALGEBRA OF INVARIANTS [CH. X and therefore H^m,f=\{pp)Ka^\ H + mj= a^' { - i (apYpa,' + 1 (ppf aj'}. Again, since we obtain as before H+mJ=-2<f>% where ^ is a quadratic. One of the factors of <f> must be a^., and if the other is ^x, then The value of t is then seen to be t = '^,{ppfax'^x. 174. We shall now briefly explain another interesting method of representing invariant properties of binary forms geometrically. If we put Xi = z and x.2 = 1 throughout the work on binary quantics the general linear substitution may be written az + b . X-, z = — -, : , smce z = — . cz + a x^ Now put z =x + iy, and represent z as the real point x, y in the Argand diagram in the usual way ; then the substitution az' + h cz -\-d is a point transformation. Unless c = the relation between z and z' may be reduced to the form {z - a) {z' - a') = k, wherein a, a', h are constants. Suppose z, z', a, a' are the points P, P', A, A' respectively, then the geometrical meaning of the above is ^P.^'P' = mod. (A;), and the sum of the angles that AP and A'P' make with any fixed line is constant. Hence the general linear substitution is equivalent to an inversion together with a change of origin and a reflexion of inclination of the line AP with respect to a fixed line. If c = the equation can be written /-^^miz-^), 173-174 a] geometrical interpretation of binary forms 209 indicating that P' is derived from P by turning BP through a fixed angle and increasing it in a given ratio, B being the point which represents yS. Hence a binary form of order n represents n real points A in the plane, and a covariant of the form represents a group of points G whose relation to the points A is unaltered by a geometrical transformation of the types indicated. In particular, the relation of the points C to the points A is unaltered by any inversion, because in the particular case in which A and A' coincide and k is real, the transformation is equivalent to an inversion with respect to A, and a reflexion with respect to the real axis through A ; but the properties of the derived figure are evidently unaltered by a reflexion alone, and hence they are unaltered by an inversion alone. 174 A. Some of the simpler invariants and covariants can now be interpreted. If az' + 1hz + c = 0; a'z^ + 2h'z + c' = represent the points A, B and G, D respectively, then when ac' + a'c - 2bb' = A, B, G, D are four harmonic points on a circle. In fact on changing the origin to 0, the middle point of AB, the first quadratic becomes z---k-=- 0, and if the second be {z — z^) (z — z^ = 0, then since the relation is invariantive we have therefore OG.OD^ OA' = 0& and OG, OD are equally inclined to OA. If we produce GO to D' making OB' = OD, we have OG.OD'=OA'=OA.OB; therefore GAD'B are coney clic, and by symmetry B is on the circle. Further as the pencil D' [AGBB] is harmonic the four points AB and GD form two harmonic pairs on the circle. We shall call them harmonically concyclic. G. & y. 14 210 THE ALGEBRA OF INVARIANTS [CH. X Ex. Shew that there is one pair of points (P, P') harmonically concyclic to each of two given pairs {A, B) and ((7, D). Further if {Q, Q') be harmonically concyclic to {A, C), {B, D) and {R, R) to {A, D), {B, C), then any two of the three pairs P, P \ Q, Q" ; R, R' are harmonically concyclic, 175. We shall now apply the complex variable to prove certain properties of the foci of conies. If the tangential equation of a conic be AP + ^Hlm + Brri' + 2GI + 2Fm + C=0, the axes of coordinates being rectangular, and x^yi be a focus, then the line a; + yi = a;i + y^t, touches the curve. Hence the above equation is satisfied by 1 = , m = , and Zi Zi we have A + -zHl- B -2 {G ■\- Fi) z^ + Cz^^ = 0. Consequently the two real foci Zi and z^ are given by the quadratic Cz'-2(G + Fc)z + {A-B + 2Ht) = 0. z +z G F Since -^—^ = n + ''n> we see that the centre is the point 2 G F C C Again, if be the origin and ^i, S^ the foci, we have SO that the origin can only be a focus when A= B and jET = 0. If a system of conies be inscribed in the same quadrilateral, their tangential equations are of the form and thus the real foci are given by the pencil of quadratics All the quadratics are harmonic to the Jacobian (/, /'), and accordingly we have the theorem that the real foci of any conic 174-176] GEOMETRICAL INTERPRETATION OF BINARY FORMS 211 inscribed in a quadrilateral are harmonically concyclic to a fixed pair of points J^, J^. We leave the reader to prove that if T^, T^ be harmonically concyclic to 8^ and S.2, then the points of contact of the tangents drawn from Ti and T^ to any conic whose foci are S^, 82 lie on a circle through T^ and T2. Hence if tangents T^P^, T^Q^ be drawn to any one of these conies, the circles T^P^Q^ all pass through another fixed point. Thus the points Ji, J^ are such that if tangents be drawn from them to any confocal to a conic inscribed in the quadrilateral, then the points of contact lie on a circle through Jj and J^. 176. The binary cubic. Suppose the form is az^ + ^hz' + ^cz -\-d = 0, and that the points A, B, G representing it are Zi, z^, z^. The cubic covariant ^ represents three points A\ B', C on the circle ABC such that A, A' are harmonically conjugate to B, G and so on. For these three points must be represented by a cubic covariant which is not /and therefore must be <}>. We have next to interpret the Hessian. Let Hi, Hnhe the points representing h^ andAj the roots of the Hessian, then we know that for real variables the range {ABGHi] is equianharmonic, (zi - Z2) (zs - hi) _ (^2 - -gs) (zi - hi) _ (zs - Z i) (Z2 - hi) I.e. :j — — . Hence since mod (w) = 1, we have on equating moduli AB . GHi = BG. AHi = GA . BHi , therefore the points Hi , H.^ are the points whose distances from the vertices are inversely proportional to the opposite sides. To construct them we draw a circle having BG for inverse points, and passing through A, with analogous circles for GA and AB; then these three circles meet in the points Hi and i/g- 14—2 212 THE ALGEBRA OF INVARIANTS [CH. X It follows in addition that H^ and H^ are inverse points with respect to the circle ABC. It is interesting to notice that the Hessian points of A'B'C are the same as those of ABC, a fact which gives rise to a curious geometrical theorem. Another easily proved property of the Hessian points is that if H^L, HiM, HiN be drawn perpendicular to the sides, then the triangle LMN'va equilateral. CHAPTER XI. APOLARITY AND ELEMENTARY GEOMETRY OF RATIONAL CURVES. 177. Two binary forms of the same order are said to be apolar when the joint invariant which is linear in the coefficients of both is zero. Suppose that the two forms in question are j> = boX-J" + nb^x^'^-^x^ + . . . + hnXa"" = bx"", then the only lineo-linear invariant is (aby = Uobn - naj)n-i +...+(- l)"an^o and this vanishes when the forms are apolar. An immediate con- sequence is that a form of odd order is always apolar to itself. Thus the discussion of apolar forms may be regarded as the development of the theory of the simplest type of invariant ; the fact that each set of coefficients only occur to the first degree in the invariant renders such a discussion simple and accounts for the relative importance of the allied geometrical properties. If two quadratic forms are apolar they are harmonic so that we may regard apolarity as being, in a certain way, the generalisation of harmonic properties. The condition for apolarity may be written in other useful forms. In fact, if the linear factors of <f> are /3x^^\ ^x^^\ ••• /3a;''*', we have = [ax\ ^x''' ^*<'' . . . y8^<">}" = (a/S"') (a/3<^0 . . . (a/3<»>). 214 THE ALGEBRA OF INVARIANTS [CH. XI Again, if <f) vanishes for the values a?i = S/i""', a^a = ya*""*, r = 1, 2, ... w, we have as far as ratios are concerned and hence the condition is This form of the condition at once shews the connection with polar forms ; given the form/, m — 1 of the vanishing points of <^ can be chosen arbitrarily and the remaining one is given by polarizing/ with respect to each of the n — 1 given values successively and equating the final result to zero. We can at once find all perfect nth. powers apolarto/, for if all the y's are the same we have so that y must be a vanishing point of/ Hence / is apolar to the nth power of any one of its linear factors, and these determine the only 7ith powers apolar to/ If/ be apolar to each of the forms ^i, 02> ••• ^m it is apolar to ^1^1 + ^^2 + . . . + ^jc^k, where the Vs are any constants ; for {/>^<^i + ^A+... + Xft<^;fc^=X,(/0i)'^+X2(/<^2r + ...+>^)t(/0A)"=O, which establishes the result. This result also follows at once from the fact that the equations of condition are all linear. Again if/ be apolar to each of the (n + 1) forms then on elimination of the a's from the equations which they satisfy, we find that the determinant of the coefficients of the ^'s is zero ; but this is precisely the condition that there should be an identical relation of the form Xi<^i4- X2^2+ ... + X„+i^,i+i = (A), and hence being given n linearly independent forms apolar to / any other apolar form is a linear combination of these. It is easy to prove directly that n linearly independent forms can be found which are apolar to a given form of order n, and this fact will 177-178] APOLARITY 215 appear independently from the special system of apolar forms to be constructed in the next article. 178. To determine n linearly independent forms apolar to a given one. I. Suppose that the factors of the given form / are all differ- ent and that except for numerical multiples they are ai'^-'o?! + a2"*>a;2, r = l,2,...n, then the nth. power of each factor is an apolar form, and they are linearly independent. For if there be a relation of the type 1 where A,,, is a constant, then the (n + 1) determinants of the array I ai<''>'\ ai^''^''-W\ «! '''''^-'Oa <»•>,... a2<''>" | r = 1, 2, ... n, vanish identically. Hence the determinant j^^(i)n aj(i)«-Ja2<i), aiW"-2o2<>>S ...tta (i)n {n)n „ (n)n—i„ (n) ~ {n)n—2„ (?i)2 (n)n Pi\ Pi''^, Px^'^pi, '--P^ vanishes for all values of pi and p,^. But this determinant being homogeneous in each set of sym- bols is equal to T—n ± n (ai<'->a2<*' ~ ai^^'ao**-') n (ai<>2 - a2"''/)i), r=l where in the first product r and s have all the values 1, 2, ...n, but are different. Thus if we choose pi,p2 such that -4= —j^v , r=l, 2, ... n, the P^ ^ determinant can only vanish when for some pair of values of r and s ai"*'a2 "'=02 "*'«!'**, in which case the two factors d^^'^'^x-^-^- ol.}'^''x^ and ai'*>a7i + a2'*'ir2 only differ by a numerical multiplier, contrary to the hypothesis that the factors are all different. 216 THE ALGEBRA OF INVARIANTS [CH. XI Consequently the n forms («!<'•' a;i + 02 ''•'a;^)", r = l,2,...n, are linearly independent when the factors of/ are all different. We are thus led at once to the interesting result that the necessary and sufficient condition that a form ^ can be represented as a linear combination of the nth powers of the factors of y is that the two forms should be apolar — the condition is necessary, be- cause as each nth power is apolar toy a linear combination of them is also, and it is sufficient, because the wth powers are linearly independent in virtue of the foregoing, supposing always that / has no multiple factors. Hence reciprocally, if <f> have no multiple factors, /can be ex- pressed as a linear combination of the nth powers of the factors of <f). This may be regarded as the extension of the elementary theorem that all quadratics harmonic to aoc^ + 2hx + c are of the form \{x — af + fi{x — yS)^ where a, y8 are the roots of ax^ + Ihx + c = 0. II. We have still to construct the apolar system when / has multiple factors. Suppose that /has the factor {ot^Xy^ + a^^'^ replacing r different linear factors, and that <^ is apolar to /. Since the relation is invariantive, we may change the variables so that the multiple factor is simply x/, and then «o> c^i> cb^y" dr-i all vanish. Hence recalling the condition ttobn - na^bn-i + ...+( — l)"an&o = 0, we see that it is satisfied for any values of ^n > Oji_i , . . . On— r+i f provided that all the other 6's vanish. Thus/ is now apolar to any form for which ho,bi,b2, ...bn-r 178] APOLARITY 217 are all zero, that is to any form which contains the factor x^^~^^'^. Among such we have the r forms ~ n— r+i/y. J — 1 ^ n—r+ir- »"— 2/y. /y. n—r-v\rr r—\ tlUn cCl t M/o *^\ 2 9 * * * 2 2 i which are obviously linearly independent. Hence, in general, when a form has a linear factor of multi- plicity r it is apolar to any form containing that factor (n — r + 1) times, and among these apolar forms it is possible to choose r which are linearly independent. If each multiple factor be treated in the same way we obtain in all n apolar forms, viz. r from each factor of multiplicity r; those derived from the same factor are linearly independent ; it remains to shew that all the n forms are so independent. Thus for the sextic x^^ {x-^ + x^- x^ we have six apolar forms in three sets Xi, x^x^, x^x^ containing the factor x-^*, (a?! + x^^x^, {xi + x^^x^ containing the factor {x^ + x^^, and x^ derived from the single factor x^. The forms in each set are linearly independent, but it has not been shewn that the different sets are independent inter se. The general proof that the sets so derived are independent presents no difficulty but is rather tedious owing to the complicated notation. (See Appendix II.) Let /= A {xi + ttiiCa)*'' (a-'i + aaa^j)'*^ . . . (ajj + cLj^^'^'p , the a's being all different, r^ + r^-^- ... + rp = n, and A a constant. This assumes that in / the coefficient of x-^^ is not zero, if it be zero we can transform /by a linear substitution into one in which x^ actually occurs. A form apolar toyis and ^-^ -^' ^—^-^ are all apolar forms since each involves the factor {x^ + aia;2)"~'*'''"^ We shall prove that this set and the corresponding ones derived from the other factors are linearly independent. 218 THE ALGEBRA OF INVARIANTS [CH. XI In fact if they are not independent the determinant /o(ai), /i(ai), ••• /n(«i) foiq), fiiql '■■fniq) where in general /« (a^) = a/ and /«'(«'-)=a^' must vanish for all values of q. Except for a non-vanishing numerical multiplier the above determinant is the limit of /o(ai), • .•/n(«0 .fn(ai + t) /o(ai + ri- ■It), Ma, + n- ~it),. -■fn{ai + n- It) Mq), -10, /l(ap + »>- -It),. '■fniOp + f'P -fniq) -It) ^ ^ (l + 2 + ... + r, -l) + (l + 2+ ... +r2-l) + ...(l + 2+ ... +rp -1) when t=0. But the determinant last written is equal to ± Jl{af, + pt-a^-a't), xll{a^ + ^'t-q), where in the first product /a = 1, 2, ... p, p' = 0,l,2, ...r,-l, o-=l,2,...p, o-' = 0,l,2,...r<,-l, except that one of the inequalities p^a, p =^a-' must be satisfied, and in the second product «r = 1, 2, .,.p. 178-179] APOLARITY 219 A linear factor is a multiple of t when p = a-, hence t occurs as a factor of the determinant to a power equal to p /3=1,2, ...p, and this is precisely the power of t in the denominator. To find the limit we put t = when p=^ a- and take the multi- plier of t in the remaining factors. Besides numerical factors which are certainly not zero, the limit is the product of a number of factors of the types ttp — a^- and Op — q, and in fact it is easily seen to be NU{ap- a^)W<r n (ttp - qYp , P+ o- P where N is an integer. Now the quantities ttj, a^, ... a.p are all unequal and we can choose q to be different from each of them, hence the determinant does not vanish for all'values of q, and consequently the n apolar forms are linearly independent. Ex. Evaluate the determinant a\ o-\ a\ a^ a, 1 ba\ 4q3, Za\ 2a, 1, 20a3, \2a\ 6a, 2, 0, /3^ ^\ ^^ ^\ i3, 1 5^S 4^3, 3/32, 2/3, 1, f. y\ ■f, r^ y» 1 179. Forms apolar to two or more given forms. Suppose we have s linearly independent forms fr = ao<'''^i" + na^ ^^^x^''-^x.^ + . . . + ««<*■* a?/*, r = 1, 2, . . . s, then the determinants of the array formed by the coefficients cannot all vanish, because in that case values of \ which satisfy s — 1 of the (n + 1) equations \ap <!' + \.^ ap^-^ + ...+ \sap '«' = would satisfy all these equations, and hence which is contrary to hypothesis. 220 THE ALGEBRA OF INVARIANTS [CH. XI If &o^i"' + nbia;j'^~^X2 + . . . + &n^2" be a form apolar to each of the fs, then r=l,2,...s. These s equations connecting the b's can be solved for s of the 6's in terms of the others, because, as we have seen, not all the deter- minants of s columns formed by the coefficients are zero. Having solved the equations we obtain s of the b's expressed as linear functions of the others, and as the remaining {n — s +1) b's are arbitrary, the general form apolar to all the /'s involves (n — s + 1) constants linearly, i.e. there are exactly (n — s + 1) linearly in- dependent forms apolar to each of s given forms. In particular it follows that there is a unique form apolar to each of n given linearly independent forms. Further if be (n — s + 1) linearly independent forms apolar to the /'s, it is clear that every <^ is apolar to every/, and that the most general form apolar to each of the <f)'s is a linear combination of the fs ; thus the relation between the two sets of forms is a reciprocal one. Some interesting results follow from this reasoning. For example, given three independent cubics, there is one form apolar to each of them, and, if its factors be all different, each of the given cubics can be expressed as a linear combination of the same three cubes, viz. the cubes of the factors of the apolar cubic. A like result applies to forms of any order and constitutes the generalisation of the problem of expressing two quadratics each as linear combinations of the same two squares. The form apolar to the three cubics aj, bi, ci, is (6c) (ca) (a6) a^ b^ c^, for this is not zero if the cubics are linearly independent and it is apolar to d^^ if (be) (ca) (ab) (ad) (bd) (cd) = 0, 179-180] APOLARITY 221 or, as can be readily seen, if ao «! tta as bo h h. h Co Ci C.2 C3 do d^ d. da = 0, which is certainly true if dx^ is the same as any of the three original cubics. The reader may verify that the equation of the apolar form may be written CtQwi t" vZjtA/2j (Xjt/'j *t" Ct2^2> 1*2^1 "i ^3^2 60^1 + &1^2> ^1^1 + ^2^2, ^2^1 + &3«2 = 0, which can be easily done by expressing this determinant sym- bolically. The extension of these results to n forms of order n will pre- sent no difficulty. 180. We may illustrate some of the foregoing results and anticipate some of the developments to come by reference to the geometry of the rational plane cubic curve. Let |, 7], ^ be homogeneous coordinates, then for all points on the curve they are rational integral functions of one parameter t and of the third order. To apply our results more directly we shall replace t by two variables which occur homogeneously, so that we have pr] = h^x-i^ + ^h^x^x^ 4- 2h^^x^ -\- h^i = /2 U and to find the point equation of the curve we must eliminate x^ and x.^ so as to obtain a result homogeneous in f, t), ^. But the properties of the curve are naturally more easily obtained by using the parametric expressions. We remark in the first place that the cubics yi,y2,/3 must be linearly independent, otherwise all such points ^, 7], ^ lie on a straight line ; next the points in which the line l^ + mrj -t- n^ = 222 THE ALGEBRA OF INVARIANTS [CH. XI meets the curve are given by the cubic which deteraaines their parameters ; hence any straight line meets the curve in three points and the curve is therefore of the third order. Now there is a unique cubic cj) apolar to fi,f2,j3, and (f> is apolar to any cubic giving the parameters of three collinear points. Conversely if a cubic be apolar to <f) the three points whose parameters are determined by it are collinear because it is of the form If I + m/a + nf. Thus the three points whose parameters are (x^, x^), (3/1, 3/2), (^i, Z2) are in a straight line if 181. Points of Inflexion. At a point of inflexion three successive points on the curve are in a straight line, and hence the parameters of the points of inflexion are determined by the perfect cubes apolar to ^, that is by <^ = 0. Hence there are at most three points of inflexion, and, since a cubic is apolar to itself, when there are three they are collinear. But there are other singularities for which three consecutive points on the curve are collinear, e.g. cusps, and accordingly we shall examine the equation </> = a little more closely. If Oa;, /8a!. Ya; be the linear factors of ^ we have with similar expressions for rj and ^ ; hence on replacing ^, r), ^ by suitable linear combinations — which is tantamount to changing the triangle of reference — we shall have from which it is readily seen that the straight lines ^=0, i; = 0, f = are inflexional tangents, and therefore in this case there are three distinct points of inflexion. If ^i = has a double factor ^^ we have I = \a^' +/i/3a.' + v^ix^ etc., 180-182] APOLARITY 223 and hence by a similar transformation we can reduce ^, r), ^ to the forms I = aa?, V = ^x, f = PxW In the neighbourhood of the point 77 = 0, ^ = 0, whose parameter is given by ^^ = 0, we see that rj- x ^, and hence this point is a cusp, while the line ^ = is an inflexional tangent. The case in which ^ = has a treble factor may be rejected because under these conditions fi,f2,fz have a common factor and, as will be readily seen, the curve breaks up into a straight line and a conic. We shall confine the further discussion to the case in which the factors of <^ are distinct. 182. Double point. To each value of the ratio x^ : x^ corre- sponds a point on the curve. The same ratio cannot give rise to two different points, but the same point may be obtained from two different values of the ratio and then it will be a double point on the curve, because every straight line through it meets the curve in only one other point. We might find the double points directly by developing this idea, but the search is best conducted in a dif- ferent manner. If X, y, z be the parameters of three coUinear points we have </>« </>2/ 0z = 0, and in general this determines z uniquely when x and y are given. When X and y give rise to the same point, and only then, the above condition does not determine z but is satisfied for all values bi z. Now the equation ^j, (py (})z = indicates that the quadratic whose vanishing points are x, y is apolar to the first polar of z with respect to ^a;^. Hence if x, y be the parameters of a double point the quadratic giving them must be apolar to all first polars of (f)^^. Two such polars are <f>x'<l>a , ^'x^fi 224 THE ALGEBRA OF INVARIANTS [CH. XI and the quadratic apolar to each of these is their Jacobian, i.e. {<t>^') ^x (fi'x <j>a<f>'fi = — {i><f>') <}>x <f>'x ^'a 4>fi, <j> and <f>' being equivalent symbols. This is 1 iU') 4>X<i>'x {<f>.<l>', - <^'a0^) = i (HI <f>x^'x iH') («/^) and the quadratic required is therefore {^^'f 4>x<f>'x = 0, namely the Hessian of <f>. Thus a rational cubic curve has always one double point, the parameters of which are given by the Hessian of the cubic giving the parameters of the points of inflexion. It will be noticed how readily properties of points on the curve are expressed by means of the form <^, and this is natural since ^ being given we can write down three forms apolar to it forfi,/^,/^ and thence find the ordinary equation of the curve. As a further example, we remark that the points of contact of the tangents drawn from the point z on the curve to the curve are given by <f>x'<f>z = 0, that is to say there are two such tangents, and the parameters of the points of contact are given by the first polar of z. Hence the quadratic giving them is apolar to the quadratic giving the para- meters of the double points. Ex. (i). Prove that the cross ratio of the pencil formed by joining th« double point to four points on the curve is equal to the cross ratio of the parameters of those four points. Ex. (ii). If the parameters of the points of inflexion be given by a^ = 0, fix = 0> yx = 0> the point of contact of the remaining tangent to the curve from the first is given by hence if/ = give the points of inflexion, the cubic covariant gives the para- meters of the three points L, M, N, here indicated. Ex. (iii). Prove that the six points in which any conic meets the cubic are given by a sextic apolar to a given sextic >//■. {i\f is apolar to the squares and products /jj/jj/j.) Thence shew that ■\\r = gives the parameters of the points where a conic can be drawn having six-point contact with the curve, and that inasmuch as these points are the points of inflexion and Z, J/, iT, ■<\r is the product of/ and its cubic covariant. 182-183] APOLARITY 225 183. Apolarity of forms of different orders. In the foregoing discussion we have seen that if x and y be the vanishing points of the Hessian of a cubic y, then a^ayttz = for all values of z, so that the Hessian although only a quadratic satisfies, in a manner, the condition of apolarity to the cubic. If ax, ^x be the factors of the Hessian, we have {aa) (a/3) ax=0, that is its second transvectant with / vanishes identically (cf. § 91). Generalising this we shall call two forms f=a^, <^=h^, m>n apolar when the nth transvectant (ab^ax'^-'' vanishes identically. Two important facts follow at once from this definition. (i) The form f is apolar to any form of order n' ^ m, having <ji for factor. For if the new form be (f>-\l/ we have (f<f>i^r=i(f<f>rH'-^> since yfr is of order n — n and the right-hand side vanishes by hypothesis. A special result is that/ is apolar to any m-ic containing the factor ^. (ii) is apolar to any polar form of f whose order is n. For let tta;" ay"^-^ be the apolar form, then [ax'^ay'^-'', hx^'Y = {(^^Y ttj/"*"", and this is zero since {ahY a^~~^ vanishes identically. Cor. <^ is apolar to any polar form of / whose order is ^ n but ^ m. The proof is as above. G. & Y. 15 226 THE ALGEBRA OF INVARIANTS [CH. XI 184. The conditions of apolarity of /= (tto, a, , . . . a^J^x^ , xj^ = a^ and (f> = {bo,b^, . . . 6„]i ar^ , a;^)" =6^;" are equivalent to m — n+1 linear homogeneous relations among the a's. In fact, equating to zero the several coefficients in (aft)" aa,"*-" we have the following equations : {aby a,'"-'* = 0, {aby a^"^-^a^ =0, ... (abf ag"^" = 0. On being expressed in terms of the actual coefficients the first of these relations involves do, Oj, ... ttn, the second (hfa^, ... a„+i, and so on, the last containing din—n , Clm—^n+\ > • • • ^n > hence if, as without loss of generality we may do, we assume that none of the coefficients b are zero, these m — n + 1 relations among the as are obviously linearly independent. It follows that by means of them we can express {m — n+l) of the coefficients a linearly in terms of the remaining coefficients and thence that there are (m -f- 1) — (m — M + 1) = n linearly independent forms of order m apolar to any given form of order n less than m. 185. Construction of a linearly independent set of forms of order m(> n) apolar to a given form ^ of order n. 1. Let the factors of the given form be /Si"-'aJi + /S.Mar, = ^^c-), r = 1, 2, ... 71, and all different. « 184-185] APOLARITY 227 Then the typical form ^^^ is apolar to h^^, for and since (/86)" = 0, when yS-B is a factor of h^, the result follows at once. Next the system of forms ye,'*-)", r = l,2, ...n is linearly independent, for if there were a relation of the type on polarizing (m — n) times with respect to y, we should have which is contrary to the established fact that the system of forms ^^M", r = l, 2, ... n is linearly independent. II. Suppose next that the factors are not all different but that Then since the factor /9a; '^' for example occurs /Xi times in ^ we know that ^ is apolar to the w-ic where G is any form of order /u.i — 1. In like manner ^ is apolar to the m-ic r being any form of order yUj — 1, for and this latter is an aggregate of forms each involving the factor But since the factor /3a; '^* occurs fi^ times in j> any form involv- ing the factor vanishes identically. 15—2 228 THE ALGEBRA OF INVARIANTS [CH. XI Hence choosing any s forms 1 1, 1 2, ••• 1 s of orders /^— 1, /^ — 1. ••• /*«— 1 respectively, forms /Sa-"* '^' Ft are all apolar to ^. Next there cannot be a linear relation between them because if there were, on polarizing it m — n times with respect to y we should obtain a linear relation of the type Tt' being of order /i( — 1. This is contrary to what was proved in constructing the apolar set of order n. Now the form ^^^'^'^''^'^'Ft contains fMt arbitrary constants and so we have a form apolar to (}> involving /*! + /^2 +•••+/*« = W arbitrary constants. The coefficients of the various constants are each apolar to <^ and they are n in number. The discovery of forms of order n apolar to a form of order m (> n) is a problem quite distinct from the foregoing. Suppose / is the given form of order m, then a form of order n which is apolar to for all values of y will be apolar to f. This condition has been shewn to be necessary and it is clearly sufficient because if then (aby ay^-'' vanishes for all values of y. Hence the form (f) sought is apolar to the m — n + l forms of order n and the problem is reduced to one in apolar forms of the same order. 185-186] APOLARITY 229 If the forms just written down be linearly independent there are linearly independent forms apolar to each of them, and thus there are at least {2n — m) linearly independent forms of order n apolar to a given form of order m. There may be more owing to the subsidiary system of forms not being linearly independent and we shall discuss this question more fully in the sequel. It is clear that if ??- > ^ there is at least one form of order n apolar to /. 186. The latter theory finds its natural illustration in the problem of representing one or more given forms of order n as the sum of a number of perfect nth powers. We shall discuss the case of a single quintic at length as an example. The general cases present no difficulties — they arise for forms of special character. Suppose the quintic is /= ttoCCi^ + oa^Xi*x^ + lOa^x^xi + lOa^x^x^ + oatXiX^* + (hx^ — f, 5_ A 5_. The second polars are linear combinations of a^ ay ay "bx^ ' dxidx^ ' dx^ ' If these are linearly independent there is a unique cubic apolar to them, and being apolar to all second polars it is apolar to the quintic itself. On referring to § 179 we may write this cubic in the form ay _ay_ _ay^ dx-^dx^ ' dx-^dx^ ' ■ dx^dx^ _ay_ _ay^ ay dx-^dx^ ' dxj)x^ ' dx^ or {hcf {caf {ahf a^ h Cx , so that it is the co variant denoted by J. 230 THE ALGEBRA OF INVARIANTS [CH. XI Now suppose then /= Xa/ + fi^J' + vy^\ <^x^xlx being all diflFerent and \, /i, v numerical. If^'= a^^^ we have for an apolar quintic /= XOa;" + /i«a;*^i + v^^, or a^;^ {'px-^ + (/a^a) + r^^. If J = OL^ we have /= Xtta," + Ataa;*a7i + x'tta,^ i»l^ or tta,^ (j^aji^' + 25'a;ia72 + rx^). This exhausts the cases in which J is not identically zero. 9Y 02/ 92^ If J be identically zero the forms ^^, ^ , —^^ are not linearly independent, because if they were they would determine a unique non-zero apolar form. Lt ^ 5!/" r^=0 dac^ dx^dx^ dx^ be the relation. Then we have 3^" d'f d'f " dx^ ^dx^dx^ dx-fixi . ' and ay ay ?/!=o ■'^ dx^dx^ dxidx^ dx} and all third polars can be expressed as linear combinations of dxj^dx^' dx-fix^' If these are linearly independent they determine a unique quadratic apolar to both and being apolar to all third polars it is apolar to the quintic. Suppose this quadratic to be (i) aA, then /= Xa/ + /iySa;' ; (ii) a^\ then J = a^* {px^ + qx^) ] (iii) identically zero, then ay ay dxi^dx2 ' dxidx^ must be identical, hence in this case all third polars are identical. 186-187] APOLARITY 231 Now all fourth polars are linear combinations of »*/- ^^ (A). dx^dx^ ' dx-^dx^ 93/" 93/" But since . xL and - — ~- are identical all the fourth polars 0x^0X2 dxidx.2^ '■ are identical with either of the forms (A). Hence the fourth polar is apolar to the quintic, for being of odd order it is apolar to itself. In this case the quintic is a perfect fifth power unless the linear apolar form is zero identically, in which case the quintic is also identically zero. This completes the discussion and leaves us with six canonical forms for a quintic, viz. (i) /= \a^' + fi^^' + vy^', (ii) y = Xct^' + fi^a^'x, + v^^\ (iii) f=\(x^^ + fxa^*x^-\-va^^x^^, (iv) / = \a^' + fi^^\ (v) /= Xa^« + fia^^Xi, (vi) /=\a^^ The discussion for any other single form can be conducted in an exactly similar manner. 187. The reader will have no difficulty in applying the method explained for the quintic to any binary form ; in particular it will be easily seen that, whereas a form of odd order (2w + 1) always has at least one apolar form of order (n + l), a form of even order 2n has not an apolar form of order n unless the determinant formed by the coefficients of its ?ith derivatives with respect to x^ and x^ be zero. Thus a form of order (2n+ 1) can in general be expressed in one way as the sum of (n + 1) (2n + l)th powers, but a form of order 2n cannot be expressed as the sum of n 2wth powers unless a certain function of the coefficients — manifestly an invariant — be zero. For example, in the sextic /= ttoXi^ + ea^Xi'^x^ + . . . + tteXo^ = 0/ = 6/ = c/ = dx\ 232 THE ALGEBRA OF INVARIANTS there is an apolar cubic only when ao ttj Oa Os tti ttj ttj a4 a, da Cbi 0,6 [CH. XI cu a. a. a« = 0. The expression of this as a symbolical form is an instructive exercise. There must be a linear relation between ay s-y d'f i.e. between ay and hence referring to § 179 we must have / = (be) (ca) (ab) (ad) (bd) (cd) ay^^b^Cidd^' = 0. Interchanging the letters in ever}' 'possible way we find that I = ^^^ (be) (ca) (ab) (ad) (bd) (cd) \ a^, a^a^, a^ai, a^ />8 /)2/t /i/»2 /% 9 Vl y t/i C/2 J t/i O2 , .1/2 ^1 > ^1 ^2 J vl'iCl'2 J 1*2 And hence the condition is (bey (cay (aby (ady (bdy (cdy = 0. The invariant / is called the catalecticant and it will be easily seen that a similar symbolical expression holds for the catalecticant of any form of even order. 188. It has been shewn that when j = identically the quintic can in general be expressed as the sum of two fifth powers, and in the course of the work we found the conditions under which it can be expressed as the sum of a smaller number of fifth powers. A similar process would of course apply to any form, but we shall now give a direct answer to the question as to what is the smallest number of nth powers in terms of which a given binary n-ic can be expressed*. See Gundelfinger, Crelle, Bd. c. 413—424. 187-189] APOLARITY 233 189. If a binary w-ic can be expressed as the sum of r nth. powers it must have an apolar r-ic whose factors are all different, so in the first place we proceed to find the necessary and sufficient conditions that the form should possess an apolar r-ic. If r > — there is always at least one apolar r-ic, 71/ Suppose, then, that r :[> ^ and that there is an apolar r-ic, namely <f>. Then <f) is apolar to all derivatives of / whose order is equal to or greater than r, and since there are (r + 1) derivatives of order n — r, viz. ay ay ay these cannot be linearly independent. Hence there is an identical relation of the form ay ay ay^ dxo on differentiating this r times with respect to Xi and x^ in the (r -f- 1) different ways possible and eliminating the A-'s, we have a^y ay a-y dx^' dx^^-^dxr, ' '"' a^/a^*- a^/ d^f d^f dxi^-'dx., ' dxi^-^dx^ dx{~'^dx2 a-/ a-y av = 0, aa;/a^/' dx^^-^dx.{+^' '"' or say Gr = 0, and it is easy to see that Gr is a co variant of y. Conversely when Gr = by the well-known theorem of Wronski* there is a linear relation between ay ay ay dx/' dx/-'dx^' '" dx/' By differentiating this we obtain two independent relations between the (r + l)th derivatives, three between the (r-|-2)th * See Appendix II. 234 THE ALGEBRA OF INVARIANTS [CH. XI derivatives and in general (p + 1) between the (r + p)th de- rivatives. Now there are (r + p + 1) derivatives of the (r +^)th class and hence of these only r are linearly independent. Hence in particular there are only r linearly independent (n — r)th derivatives and as these are of order r there is one form of order r apolar to them and therefore apolar to the form f. Hence when Or=0 there is an apolar form of order r. Thus forming the successive covariants Go, Gi, G2, ..., the necessary and sufficient condition for an apolar r-ic is Gr = 0. If Gr-i 4= there is no apolar form of order less than r, for if there were any such apolar forms there would be at least one of order (r — 1) and Gr-i would vanish. Hence if Gr be the first of the covariants G which vanishes the lowest order of an apolar form is r. Finally if Gr-i 4= 0, Gr = there is only one apolar form of order r. Suppose in fact there are two apolar forms of order r and that for simplicity their factors are all different in both cases. Let (a.1 + agiPg), s=l,2,, . . ., r be the factors of the first, and (xi + ^.x^), 5=1, 2, ..., r be the factors of the second, then we have r r 1 1 therefore there is an identical relation of the type r r 1 1 Now 2r is less than w, hence by § 178 such a relation is only possible when the coefficients of the various 7ith powers severally 189-190] APOLARITY 235 vanish ; thus since the a's are all different and the yS's are all different it follows that either every X and every jjl is zero or else for a certain number t of values of s while for other values of s Consequently f can be expressed as the sum of t nth powers where t<r, hence there is an apolar form of order t. But in this case we must have Gr-i = contrary to hypothesis, hence there is only one apolar form of order r. The reader will easily establish the fact that if there are only two apolar r-ics they must have (r— 1) common factors and that these factors multiplied together give the apolar (r — l)-ic. Further the extension of the above to the case in which two or more a's are equal will present no difficulty. Cor. Since Gi is the Hessian of/ the necessary and sufficient condition that / should be a perfect nth power is that its Hessian should vanish identically. Ex. (i). If / is apolar to ^ then / is apolar to every form having ^ for factor. Ex. (ii). If a binary form of order n have an apolar r-ic {r < %), then it has at least (s + 1 ) independent apolar forms of order r + s. Ex. (iii). Shew that the argument of § 188 can be extended to any number of binary forms, and construct a table of canonical forms of a simultaneous system consisting of a cubic and a quartic. Ex. (iv). Shew that in general two forms of orders n^ and n^ can be expressed as linear combinations of powers of p linear forms if 3jo — 2 = Wi + »2 ; find the jo-ic giving these linear forms and extend the results to any number of forms. Ex. (v). From the symbolical form of a catalecticant deduce the sym- bolical forms of the covariants G. 190. We shall conclude this chapter with a few geometrical illustrations of the foregoing theory. 236 THE ALGEBRA OF INVARIANTS [CH. XI Binary Quadratics in connection with the Geometry of a Conic. If we have the equations ^ = a^Xi^ + la^XyX^ + a^x^ = a^ =/, then the point |, tj, ^ lies on a fixed conic. The equation of the conic is easy to find. For the line touches the curve when Disct. {Xf+ fi<f> + mjr) = 0, i.e. if Vt'ii + /i^*2-2 + ^*33 + 2/iM23 + ^vXigi + 2XjjLii2 = 0, where t'la = (/, <f>y etc. This being the tangential equation the point equation is *11 *12 *13 s I ~ "> 1l2 ^22 *2S V *13 ''^23 *33 b I ^7 r , or 7„ f= + 72217^ + 733 r^, + 21^7}^ + 2T^ ^1 + ^J'u^ = 0. where /,« is the minor of v« in the determinant hi *12 ^3 *ia *22 *23 *1S *2S *33 In particular the equation of the conic gives an identical relation between three quadratic forms and their invariants. 191. An immediate inference from the parametric expressions for ^, 7), ^ is that the cross-ratio of the pencil joining a variable point P on the conic to four fixed points x, y, z, o> on the curve is equal to the cross-ratio of the parameters of those four points, i.e. (xy) (zo)) {xo) (zy) ' For let the equations of two lines through P be X = 0, Y=0 and let X+tY=0 be the equation of the line joining P to the i 190-192] APOLARITY 237 point X on the curve. Then there is an algebraic relation con- necting t with -^ and since to one value of i corresponds one value of — and vice versa we must have x^ _ Ax-i + Bx2 ~ Cxi + Dx2 ' where A, B, C, D are constants. Hence the cross-ratio of four values of t is equal to the cross- ratio of the corresponding four values of the parameter — . Xt^ 192. In connection with this conic there is a simple corre- spondence between binary quadratics and straight lines in the plane, for a binary quadratic % equated to zero gives two points P, Q on the conic, so that if we make % correspond to the line PQ when either is given the other is uniquely determined. The quadratic % can be written in one way in the form \f+fi(}> + vylr, and then \^ + fir) + v^=0 is the equation of the corresponding straight line. Accordingly if the line pass through a fixed point in the plane, say |o. Vo, ?o, we have = \f+fi<f> r^ ^|r bo = '^(/- l>)+K^-|>) and 'x^ is apolar to (i.e. harmonic to) the fixed quadratic apolar to /-|Vand(^-|?t. bO bo Hence if the line PQ passes through a fixed point T, the corresponding quadratic '^^ is harmonic to a fixed quadratic t. Now the perfect squares apolar to t correspond to lines touching the conic, since in this case the points P and Q coincide, hence these perfect squares determine the points of contact of the tangents drawn from T to the conic. 238 THE ALGEBRA OF INVARIANTS [CH. XI If these points are R, S, then the quadratic giving R and S is apolar to the square of the linear forms giving R and »S respec- tively, and since J is apolar to these latter squares, it follows that J corresponds to RS, as can be seen in many ways. Thus if PQ pass through T, and jR^ be the polar of T, the quadratics corresponding to PQ, RS are harmonic, or in other words, when two quadratics are harmonic, the corresponding lines are conjugate with respect to the conic. This can be easily verified by using the tangential equation of the conic. Consider a binary quartic representing four points A, B, C, D on the conic, and let BG, AD meet in E, CA, BD in F, and AB, CD in Q. Then the quadratic corresponding to the polar of E is harmonic to the two quadratics corresponding to AD and BG, i.e. this polar meets the conic in two points which are the double points of the involution having B, G and A, D ior conjugate elements. Thus the polars of E, F, G meet the conic in the double points of the three involutions determined by the four points A, B,G,D; but EFG being a self-conjugate triangle of the conic, the polar of E is FG, and the lines EF, FG for example are conjugate lines with respect to the triangle, hence the pairs of double points of any two of the three involutions are harmonically conjugate. [This corresponds to the fact that the sextic covariant of a quartic can be written as the product of three quadratics which are mutually harmonic] 192-193] APOLARITY 239 193. As another example, let us prove that a triangle and its polar triangle with respect to the conic are in perspective. Suppose the sides of the triangle A1A.2A3 correspond to the quadratics /j, /g. fs respectively, and that the sides of the polar triangle B^B^B^ correspond to ^i, ^2. ^3 respectively. Then since jBjjBj is conjugate to A-iA^ and A^A^, it follows that <^3 is harmonic to ^ and f■^, so that </)s is the Jacobian of fx and f^. Hence the sides of B^BiB^ correspond to Jts, J 31, and Jia where J^^ = {f^f^). Now let B^B^ meet A-^A^ in Pj, then the polar of Pj is conjugate to both these lines, and therefore corresponds to (<^i,/i)orto (/i, J23). The polars of the analogous points Pg, Pg correspond to (/a, Jzi), (/s, J12) respectively, and Pj, Pg, P3 will be collinear if their polars are concurrent, i.e. if the quadratics KJly *J^), (/2, ''Sl/j \fz^ J 12) are harmonic to the same quadratic. To prove that this is so, let us calculate the quadratic harmonic to the first two. Representing J\> J2> y 3 Oy eta; > Oa; , CjB , we have J23 = (J>c) h^Cx, etc. .'. (/i, Jo^)' = [ax, (be) hcx] = i (ac) {be) aj>^ + ^ (ab) (be) a^c^ = i {{acY bx' + {bey ax' - {abf Cx'] - \ {{abf Cx' + {bcf a^^ -{acfbx'] = \{acfbx'-^{abycx'. Similarly if. , J31) = i {aby Cx' - i {bey ax', and the quadratic harmonic to these two is {{acy bx' - {aby Cx\ {aby Cx' - {bey aj^], or {aby {acy J^ - {aey {bey J^ + {aby {bey /31 , or {aby {acy J^ + {bay {bey J^ + {cay {eby J,„ and by symmetry this is harmonic to {f^, J^^ also. 240 THE ALGEBRA OF INVARIANTS [CH. XI Thus the polars of P1P2P3 meet in a point whose polar corresponds to *23 *31 *12 where 1*23 = (f^, f^f = (bey. And hence P^PJPz lie on the straight line corresponding to the quadratic 1^ I3I t>i2 The reader will find it interesting to shew that the above quadratic may be written T T T ~ ' where the I's are derived from the is as before. Ex. (i). ABC is a triangle inscribed in a conic, and the tangents at J, B, Cmeet BC, CA, AB in Z, Jf, N. Determine the parameters of the points in which the polar of L meets the conic in terms of those of A, B, C. Hence shew that Z, M, N are coUinear, and that the line LMN meets the conic in the Hessian points of ABC. Ex. (ii). Deduce Pascal's Theorem from the parametric representation of a conic. Ex. (iii). Prove that six conies can be drawn through four fixed points to touch a given conic, and that their points of contact are given by the Jacobian of the quartics determined by any two conies through the four points. 194. Twisted Cubic. The relation of the rational cubic space curve to the binary cubic is exactly the same as that of the conic to the binary quadratic. The line of argument is similar to that used in the previous case. We have 1^ = a^x^ + SajiCi^dJa + ^a^XiX^ + a^x.^ = ai =/, rj = bj' = <f), and the curve represented is a cubic because the plane X^ + fjbT] + v^ + prs = meets it in three points. 193-195] APOLARITY 241 We have now an exact correspondence between planes and binary cubics. If a plane pass through a fixed point T the corresponding cubic is apolar to a fixed cubic which determines P, Q, R the points of contact of the osculating planes of the cubic through T. Cf. § 192. Hence there are three such points, and since a cubic is apolar to itself, the plane joining them passes through T. We shall call T the pole of the plane PQR; and we observe that when two cubics are apolar, each of the corresponding planes passes through the pole of the other. If TLM be a chord of the cubic through T, then any plane through this line is apolar to the cubic giving PQR, so that L, M are determined by a quadratic apolar to the cubic, i.e. by the Hessian of the cubic. 195. We shall illustrate the above remarks by the discussion of an algebraical problem intimately connected with the cubic curve. The planes passing through a fixed line I correspond to cubics of the form where \ and X^ are variable numbers. We call such a linear system of cubics a pencil, and then we see that the Jacobian of two members of the pencil is J = (Xlf^2 — \2fll) (/1/2), so that except for a numerical factor it is the same for every pair of cubics in the pencil. Hence we may call this quartic the Jacobian of the pencil and the question arises — Given a binari/ quartic J, can a pencil of cubics be found of which it is the Jacobian ? To answer this question consider the geometrical meaning of the Jacobian with reference to the- line I. If the cubic \fi + X.2/2 have a double factor this factor occurs in the Jacobian, because it occurs in both G. & Y. 16 242 THE ALGEBRA OF INVARIANTS [CH. XI And further, as the discriminant of a cubic is of degree four there are only four members of the pencil which have a double factor; therefore the Jacobian of the forms contains these four double factors and no others. But if a cubic Xj/i + X^/a have a double factor the corresponding plane touches the curve, and hence through a given line four planes can be drawn touching the curve, and their points of contact are given by / = 0; in other words there are four tangent lines to the curve which intersect a given line I, and they are tangents at the points given by J=0. Now our problem is equivalent to the following : Being given the four points of contact, to construct the line I. But the four tangents being given there are two lines meeting them, and, as each of these corresponds to a pencil of cubics having / for Jacobian, it follows that the question is always soluble and that there are two solutions which may coincide in particular cases. Ex. (i). Prove that three osculating planes and one chord of a cubic can be drawn through any point. The chord determines the Hessian of the cubic giving the points of contact of the planes. Ex. (ii). The plane joining the points of contact of the three osculating planes from a point passes through and is called the pt)lar plane of 0. If the polar plane of passes through (Z the polar plane of (7 passes through and the corresponding cubics are apolar. Ex. (iii). By using line coordinates prove that the tangents to the cubic belong to a linear complex and that any two planes cori-esponding to apolar cubics meet in a line belonging to this complex. Ex. (iv). If a line I belong to the above complex the points of contact of the tangents to the cubic which meet I are given by a quartic for which the invariant i vanishes. In this case the two pencils of cubics having the quartic for Jacobian coincide. Ex. (v). If two pencils of cubics have the same Jacobian, then any member of the first i)encil is apolar to any member of the second i^eucil. 196. The twisted quartic. The rational space curve of the fourth degree furnishes the most convenient geometrical representation of a binary quartic and its concomitants. Suppose that ^ = Px\ V = qx\ K^'Tx, ^ = t,* 195-197] APOLARITY 243 is the parametric representation of such a curve. Then since ^, 7j, f, zT are not connected by a linear relation the four binary quartics are linearly independent and are therefore all apolar to a unique quarticy which we shall denote by Ux — ^x — ^x — ^x • It is evident that four points on the curve are coplanar when and only when the quartic determining their parameters is apolar to f — in fact such quartics are linear combinations of f, i], ^, tsr. Hence the four points a, /3, 7, 8 are coplanar when aattpaytts = 0. Thus ttx* = gives the four points of superosculation, i.e. the four points at which the osculating plane contains four consecutive points of the curve. Through a point 8 on the curve can be drawn three osculating planes other than that at B and their points of contact are given by cix'^cis = 0, i.e. by the first polar of B. By varying B we obtain a pencil of cubics each of which determines three points on the curve such that the osculating planes at them meet in a point on the curve. Now if a member of this pencil have a double factor we have two consecutive osculating planes whose line of intersection meets the curve, and hence the double factor gives a point the tangent at Avhich meets the curve again. But the double factors of members of the pencil are precisely the factors of the Jacobian of the pencil, which in this case is the Hessian of f. Hence 11 = gives four points on the curve the tangents at which meet the curve again. 197. We have still to interpret the sextic covariant which gives six points on the curve. For this purpose let us seek for points P, Q on the curve such that the osculating plane at P passes through Q and the osculating plane at Q passes through P. If X, /i, are the parameters of P and Q we have a\^a^ = 0, 16—2 244 THE ALGEBRA OF INVARIANTS [CH. XI To eliminate \ we note that from the second equation and as \ occurs to degree three in the first equation we must use three different equations of this type. Hence we have (ah) (ac) (ad) a^h^^c^^d^^ = as the equation for fjb. Now (ab) (ac) (ad) axhiCx^d^^ = \ (ab) aj)^c^d^ {(acj d^' + (adf c^ - (cdj a^] = \ (ab) (acf a^bic^ •d^-\-\ (ab) (adf axbid^- . c^ -\(ab)a^^bi.(cd)''CxH^ where t is the sextic covariant. Fory=0 the points P and Q coincide, hence there are three pairs of points such that each lies in the osculating plane of the other and they are given by the sextic covariant. 198. We shall now sketch some further investigations con- nected with the curve. The triple secants of the curve are given by the pencil of cubics apolar to/, so that if one cubic give three points on a triple secant and another three points whose osculating planes meet on the curve the two cubics are apolar. The two pencils have the same Jacobian, as follows from geometry and analysis. The four points in which the tangents at the Hessian points meet the curve again are given by (ahY ((nc) (bd) (cdy ajy^c^d^ = 0, for the discriminant of their first polars must vanish. This reduces to iH —jf= 0. If t = the four points of superosculation are coplanar. If j = there is an actual double point on the curve, because in that case / has an apolar quadratic — the quartic is now the complete intersection of two quadrics which touch, whereas in general it is the partial intersection of a quadric and a cubic. 197-198] APOLARITY 245 There are two other tangents meeting any given tangent to the curve and when j = there exist pairs of chords such that the tangents at the extremities of either intersect those at the extremities of the other. Between the parameters of the points of contact of two intersecting tangents there is a symmetrical (2, 2) relation, hence by comparison with Poncelet's porism we infer that if one twisted polygon with n sides can be circumscribed to the curve there is an infinite number of such circumscribing polygons. By forming the expressions for the six coordinates of the tangent at any point, which are the Jacobians of the fundamental quartics taken in pairs, we see that the condition that six tangents should belong to the same linear complex is that the sextic determining their point of contact should be apolar to a fixed sextic. This fixed sextic can be shewn to be the sextic covariant either geometrically or analytically. We can in fact establish the following theorem : If two quartics are apolar to a third then their Jacobian is apolar to the sextic covariant of the latter. Ex. (ii), p. 52. The sextic covariant therefore bears the same relation to the line geometry of the curve as the fundamental quartic does to the point geometry. By discussion of the quartics apolar to this sextic the reader will easily prove that the tangents at four points given by a quartic covariant (of the type Xilf + fijf) are generators of a quadric. References to various memoirs of Reye, Rosanes and others on the subjects discussed in this chapter will be found in Meyer's Berichte. See also the same author's book Apolaritat und rationale Curve. There is an interesting paper on the twisted cubic by Sturm in Crelle^s Journal, Vol. Lxxxvi., and a very full list of references for the twisted quartic in a paper by Richmond, Camh. Phil. Trans. 1900. CHAPTER XII. TERNARY FORMS. 199. In this chapter we shall extend the symbolical notation, which has been used throughout for binary forms, to the case of forms with a higher number of variables. It may be well to remark at the outset that the methods developed up to the present are nowhere else so effective as in the case of binary forms; accordingly, while we shall explain at some length the notation and methods for ternary forms, we shall content our- selves with a very brief indication of the extension to forms with four or more variables. 200. A ternary form in one set of variables x^, x^, x^ will be written in the form / = S , .^., apqr x,PxM/ ; p + q + r=n, and the summation is of course extended to all possible values of the integers p, q, r satisfying this condition. In agreement with the symbols previously introduced, we shall represent y* by the umbral expression so that a-^PaSoL^ — apqr. Just as before an expression in the a's has no actual (as opposed to symbolical) significance unless the total degree in the a's is exactly n. To represent forms whose degree in the a's is greater than unity, we have to introduce other equivalent systems of symbols /3 and 7, so that J — "a; Hx — v* • • • 199-201] TERNARY FORMS 247 201. Let us at once point out how concisely polar forms are represented in this notation ; the rth polar of yiy^ys with respect to / is in-r)\{ d d dy. the numerical factor being introduced for arithmetical conve- nience. This expression is {n-r)\( d_ d_ " dx^^y^bx, ' ^'dx.. (n — r) ! n ! n ! {n — r) y^^+y^uz+y^T^^) («i^^ + OsaJj + agajs)'* a "'-''a »■ i.e. the rth polar is represented by Thus for example the ecjuation «/ = represents a curve of order n, and if 3/ be a point on it the equation of the tangent at that point is axOy''-' = 0. Again, the points of contact of the tangents drawn from a point y to the curve lie on the first polar Next let us consider the effect of a linear substitution on a form such as f. We shall write such a transformation in the form ^1 = ll^l + »7l^2 + fl^S \ «2 = la-^Ti + 7/2X2 + ?2^3 >• X3 = ^jZi + V3X.2 + ^3^3 J The effect of this change of variables on olx is to change it into ai(|i^i+^iA^+ ^,X,)+u.,{^,X\ + rj,X,+^,Xs) + a,(^,X,-^v3X,+ ^,X,), or a^Xi + a^X., + a^Xj . 248 THE ALGEBRA OF INVARIANTS [CH. XII Hence / which is cn^^ becomes or S , ; , a^a'ia/XyPXsX/, where « + g + r = n. Thus the coefficient of Zi" is a/*^ or the result of putting the l^'s for the ip's in/, and subsequent coefficients may be deduced from this by means of the polarizing operators operating on the first. 202. The above will shew how very convenient the symbo- lical notation is for dealing with constantly occurring functions like polars, but, as in the case of binary forms, its great value lies in the application to the theory of invariants and covariants. Let us recall the significance of these terms especially for ternary forms and illustrate them by some examples. Suppose that after the application of a linear transformation Xr = Ir^i + rj^X^ + ^r^s. ^ = 1, 2, 3, a ternary quantic p\q\r f- ^^w \^\„\„\ ^^i^^aW becomes F = 2^„„^ , ; , XfX^X^, ^ 'p\q\r\ then a rational integral function / of the coefficients is said to be an invariant when where M depends only on the transformation. A covariant of / possesses a similar property, but involves the variables as well as the coefficients, and similar definitions apply to invariants and covariants of two or more forms. After what has been said on the subject in dealing with binary forms, we need not stop to prove that it suffices to consider invariants and covariants which are homogeneous in the coefficients of each form involved, and covariants which are homogeneous in the variables — further it may be easily proved 201-203] TERNARY FORMS 249 that the factor M referred to in the definition must be a power of the determinant of the transformation, and we shall assume henceforth that it is so. The reader may regard this as a simplifi- cation of the definition, or supply the necessary proof on the lines of that given for binary forms. See § 24. As examples, we may notice that the discriminant ahc + %fgh — af^ — hg- — ch^ is an invariant of the ternary quadric awi^ + hx2 + cxz + ^fx^s + ^g^^i + 2hxiX.2. In this case the power of the determinant which occurs as multiplying factor is two. Again the Hessian. BY dxi^ dx^dxi BY ay dxidx2 dx2 ay jy dx-^dxs ey dx^dx^ ay dx^- dx^dxi dxs dx2 is a covariant of any ternary form /. As a further example, the Jacobian of three forms is a covariant of the three forms. We shall not stop to verify that these expressions actually are invariants or covariants as the case may be. They are well known in the theory of curves and we mention them here to remind the reader that such invariant functions as defined above do actually exist. By the use of the symbolical notation we shall be able to verify our assertions much more easily, and also to construct any number of invariants and covariants. 203. Just as in the theory of binary forms we denoted the determinant I «! a., by (a^), I A A so now we shall denote «! Oo a, I by (a^y). /3x /3. /33 j 7i 72 73 i 250 THE ALGEBRA OF INVARIANTS [CH. XII We are now in a position to enunciate and prove the first theorem connecting invariants and symbolical notation, namely : Every expression represented symbolically by factors of the type (a^y) is an invariant, and every expression represented by factors of the types (a^y) and a^ is a covaHant. For by a linear substitution a^ becomes and hence {(x&y) becomes i.e. (a/37)(|770. Qf a, a^ A ^r, ^C 7^ yv 7f So that the expression formed from the new coefficients is equal to that formed from the old coefficients multiplied by a power of (^r/^) ; and the power of (^v^) that occurs is equal to the number of symbolical factors of the type (a^^Y) that occur in the expression, 204 The preceding proof, depending only on the way in which the symbolical letters occur in the expression under consideration, applies equally well to an invariant or covariant of any number of ternary forms. The only further condition that such a symbolical expression should actually represent a covariant is that it should have a meaning when expressed in terms of the original coefficients, i.e. each symbolical letter must occur to the requisite degree in every term in which it appears. It is easy to verify that the three examples already given are actually invariants or covariants as the case may be. Thus if Oaoo^i'^ + ao!!o^2^ + aoo-2^3^ + ^ani/ViXz + 2aioia;3^i + 2aou«'3p^3 = Ctx- = ^x' = yx, we have by direct multiplication (a/87)' = (2±oA73)^ = 6 ^200* ^110 > C^lOl ^110 > Cto20> ftoJl ttioi» ^011 J Ct002 since a.-^ = fii^ = y^ = a^oo , etc. Thus the discriminant of the ternary quadric is an invariant. 203-205] TERNARY FORMS Again, if we have three forms then 3-1= aia^".-i, a,aa;"i-i, tta"^"'-^ A/3^'^^-\ /3./8^'^-\ Mx''^-' 7i7/''"'. 727*"'"S 7a7x"=-^ 1 3 8^1 9/i 8a?2 5/i ^ A SC/iA/a) njU^n n^natig 8 {x l*'^2^3^ a/; 9a;2 3^3 dx.2 which shews at on ice that the Jacobia n is a CO variant. 251 The fact that (ot^yf ax^'-^^x'^'^Jx''^ represents the Hessian of the form /= a^'^ = /3a;" = jx^ niay be easily verified in the same way. The deduction of the equation of the Hessian from the property that the polar conic of any point on it degenerates into two straight lines affords an instructive example of the symbolical calculus. The polar conic of the point yi, y^, y-i with respect to the curve /= Oix'' = ^x'' = 7x" = is represented by the equation and the discriminant of this is Hence changing y into x we get the Hessian as the locus of points possessing the property in question. 205. Geometrical Meaning of a Linear Transformation. To obtain a geometrical representation of ternary forms we naturally suppose the variables x^, x^, x^ to be the homogeneous coordinates of a point in a plane referred to a certain triangle. The equation to zero of a ternary form will then represent a curve in the plane. A linear transformation Xr = t,X, + rj,X, + t:,X,, r = 1, 2, 3, may be regarded from two points of view, for if P be the point 252 THE ALGEBRA OF INVARIANTS [CH. XII sCi, x.i,Xz we may either suppose P to be unaltered and the triangle of reference changed or we may suppose P to be changed and the triangle of reference unaltered, in other words X^^X^, X, may be the coordinates of the original point referred to a new fundamental triangle or they may be the coordinates of a new point (into which P is changed) referred to the original triangle of reference. The general linear transformation as written above can be, in fact, represented as a change in the triangle of reference together with a multiplication of each coordinate by a suitable quantity — the equations of the sides of the old triangle of reference referred to the new triangle are Xr = 0, that is ^rX^ + 77rXj + frXs = 0, r = 1, 2, 3, and on solving for X^, X^, X^ in terms of x^, Xo, x^ we can easily find the equations of the sides of the new triangle of reference in the old coordinates. From this point of view, if I be an invariant of a ternary form f, when / vanishes for one triangle of reference it vanishes for any other triangle of reference ; that is to say, 7=0 expresses a property of the curve itself and not a relation between the curve and some particular triangle. In like manner the curve represented by a covariant is con- nected with the original curve by relations which are quite independent of the triangle of reference. From the second point of view the point P is changed into a point P' by a homographic transformation, i.e. a transformation possessing the following properties : (i) any point P is changed into a point P ; (ii) any straight line 'p is changed into a straight line /;'; (iii) if P lies on 'p then P' lies on p ; (iv) the cross-ratio of four collinear points is equal to the cross-ratio of the four corresponding points ; and the cross-ratio of four concurrent straight lines is equal to the cross-ratio of the four corresponding lines. Of these properties (iv) is the only one we need prove. Let jpi = and P2 = be the equations of two straight lines and p/ = 0, p2=0 the equations of the corresponding lines, then 205-206] TERNARY FORMS 253 Pi +\p.2 =0 corresponds to pi + \p2 = for all values of X, and inasmuch as the cross-ratio of the four lines Pi + \rP2 = 0, r = l, 2, 3, 4 IS — (X.2 — Xs ) (Xi — X4) the result follows at once. (X3-X,)(X2-X4) Again, suppose that a figure in one plane is projected into another plane and that P' is the projection of F, then if x^, x^, x^ be the coordinates of P referred to a fixed triangle in the first plane and Xj, Xo, X3 be the coordinates of P' referred to a triangle in the second plane, it is very easy to see that x^, x^, x^ are linear functions of X^, X^, X3 and vice versa. On the con- nection between projection and linear transformation see § 157. Thus projection affords an example of linear transformation and in this case the four properties enunciated above are self- evident. Hence if / be an invariant of a form /, then 7 = expresses a property of the curve /= that is unaltered by any homo- graphic transformation and in particular by projection. A like remark applies to the connection between a covariant curve and the original curve — it is undisturbed by projection. 206. The connection between these two different points of view is interesting. The linear transformation Xr = ^rXi + rjrXn + ^^Zj leaves three points of the plane unaltered in general. In fact to find the points whose position is unchanged we have merely to put Xr = px,. in the equations above and then elimination of x^, x^, x^ 1 gives 5 a cubic for - , VIZ. P Vi, V2- Vi> ^3- =0. 254 THE ALGEBRA OF INVARIANTS [CH. XII lu general this equation has three unequal roots and to each root corresponds a single determination of the ratios x^ : x^ : x^. If we take the triangle formed by these three points for triangle of reference the linear transformation takes the simple form 00^ — - aTj ,A j y U/2 ^~ A^2 "^ 2 ) ^H ^~ "^3 "^ 3 • And hence in general by a suitable choice of the triangle of reference a linear transformation can be reduced to this form in which the coordinates are only multiplied by constants. Ex. (i). Prove that by suitably choosing the coordinate system any four points i^nay be reduced to the form ^.^(i) = 0, A-3(i) = 0, .Vj(-^) = 0, A-3(2)=o, :r,(3)=0, a:p=0, And hence prove that a linear transformation can be found which changes any four points into four given points. Ex. (ii). Given four points and their four con-esponding points, shew that the point corresponding to any fifth point can be constructed by means of the ruler only. (Use the cross-ratio property.) Ex. (iii). Hence or otherwise shew that any linear transformation is equivalent to a ruler construction. Ex. (iv). If the linear transformation leave an isolated point and every point on a fixed line unaltered, shew that the equation of § 206 has a double root which is a root of every first minor. Give an equivalent geometrical con- stmction in this case. 207. Starting with the definitions of invariants and covariants already given we might proceed to prove that every invariant and covariant can be represented symbolically in the form indicated in a previous theorem, and then go on to develop a theory of invariants and covariants as far as possible on the lines of binary forms. However it is essential to remember that the real and primary importance of such a theory lies in its application to geometry, and as in geometry line coordinates and tangential equations occur quite at the beginning of the analytical ex- position, we are led here to introduce line coordinates as well as point coordinates from the first. Indeed it is not difficult to shew on purely analytical grounds that line coordinates are essential for the proper treatment of the algebraical questions that arise ; such an explanation together with a comparison with the theory of binary forms will be given later. 206-207] TERNARY FORMS 255 The equation of a straight line being UlCCi + u^x., + 113X3 = 0, Ui, U.2, W3 are as usual called the coordinates of the line. Following Clebsch it is preferable to regard the equation just written as neither the point equation of the line nor the line equation of the point, but as the condition that the line (u^, Uo, u-^ and the point (a?!, X2, X3) should bear a certain relation to one another. The first fact to notice about (wj, ti^, ih) is that they are contragredient to Xi, x.^, X3, for if Ui, u^, u^ become Ui, U.,, U3 and x^, x\2, X3 become X^, X^, X3, then u■^x■^ + u^x.2-¥ U3X3 must become U^X^-\- U^X^-\- U3X3, which is precisely the condition for contragredience (§ 39). Further, the us being contragredient to the x's are cogredient with the a's, and in fact by the linear transformation of § 201 we see at once that Wj becomes u^, it^ becomes u^ and Wg becomes u^. It is now evident that by a linear transformation the factor (a^u) merely becomes itself multiplied by the determinant of transformation, so that every form whose symbolical expression consists exclusively of factors of the types possesses the property of invariance. There are four classes of such invariant functions. (i) Containing neither x's nor us. These as already mentioned are called invariants. (ii) Containing x's but not us. These are covariants. (iii) Containing u's but not x's. These are new introductions and are called contravariants. (iv) Containing both u's and x's. These are called mixed concomitants. Further there is the identical invariant form u^ which does not contain the coeflficients of the original forms at all. The degree to which the coefficients occur in a given invariant form is called simply its degree, the degree in the x's is called its order and the degree in the u's is called its class. Chw flioi U, ^020 C^Oll M2 floii ^^002 Us ^<2 Us 256 THE ALGEBRA OF INVARIANTS [CH. XII Examples of invariants and covariants have already been given. As an example of a contravariant we may mention as a contravariant of the conic. In fact equated to zero it represents the line equation, and the reader may verify by direct multiplication that it is equivalent to (a^uy, and therefore actually is an invariant form. As an example of calculation similar to that given for the Hessian consider the locus of a point whose polar conic with respect to the cubic touches a given straight line u. The polar conic of the point 1/ is and the condition that this should touch the given line is Thus the equation of the required locus is (a/3aycL,^^ = 0. This is a mixed concomitant of the cubic. Equated to zero it represents the point equation of a curve when the lis are taken as constants, viz. the point equation of the locus mentioned. When the x's are taken to be constant it represents the line equation of a curve, viz. the line equation of the polar conic of the point w. Similar remarks apply to mixed concomitants in general. 208. Principle of Duality. The fact that the condition that a straight line whose equation is a^; = should pass through a given point is symmetrical in the coordinates of the line and the point leads to an important remark. For if the proof of any theorem relating to a figure containing m straight lines and n 207-209] TERNARY FORMS 257 points is thrown into an analytical form, then by interchanging point and line coordinates throughout the investigation we obtain a correlative theorem for n straight lines and m points — if in the first figure one of the straight lines pass through one of the points, then in the second figure the corresponding point lies on the corresponding straight line. It must be clearly understood that such theorems as are here contemplated depend only on lines passing through points and points lying on lines, so that the process is exactly analogous to the reciprocation of descriptive properties. Naturally we can pass to point loci and line envelopes by the introduction of an infinite number of points and lines into the figures. A fuller explanation of this principle of duality will be found in works on geometry, but the above will suffice as an indication of some general ideas which will underlie much of our subsequent work, especially on the theory of two conies. Ex. (i). Shew that in every case the theorem dual to a given one may be obtained by reciprocating the given one (supposed descriptive) with respect to a conic. Ex. (ii). If 2), E, F be three points in the sides BC, CA, AB of a triangle, and AD, BE, CF be concurrent, then the lines EF, FD, DE meet BC, CA, AB respectively in thi'ee coUinear points. What is the dual theorem ? 209. Methods for transforming symbolical expressions. As in the case of binary forms there are two different methods by which a symbolical expression may be transformed : (i) the interchange of equivalent symbols, (ii) the use of identities among elementary symbolical ex- pressions. As an illustration of (i) we may notice that for the ternary quadratic aj = fi^^ — etc. the mixed concomitant {a0u)ax^x vanishes identically. In fact, we have since a, /8 are equivalent and (o^w) ax^x = - i^oLu) ax/3x by the properties of determinants, hence (a/3w) ax0x = 0. G. & y. 17 258 THE ALGEBRA OF INVARIANTS [CH. XII The fundamental identity for ternary forms is (^878) a, - iySa) ^^ + (Ba^) 7, - {ccBy) 8^ = 0,' deduced from j ai /3i 7i 5i == 0. I as ^2 72 S.2 «3 ^3 73 83 «« ^X Jx Bx On replacing 8 by u (^yu) ax - (yua) ^^ + (uafi) yx = {aM "« or {^yu)ax + (yau)^x + {ai3u)yx = Wy)Ux, an easily remembered result. Again, if in the first identity on replacing x by (e^), i.e. x^, X2, X3 by €2^3-63^2, ^3^1-61^3, 61^2 -eofi respectively, we deduce («^7) (H) - (M) (yeO + (7S«) (0eO - (0y8) (a.n = 0. Another important identity is where Wj, u^, u^ are the coordinates of the line joining the points a; and y so that Ui = a^a^s - ^^s^i, W2 = a;3i/i - x^ y^, u^ = x^y.- a?22/i . In exactly the same way we have where x is the point of intersection of the lines v and w. Ex. For the ternary quadratic a^=^^=Qtc. shew that (a^u) (ayu) ^^y,=i (a/3«)2 . y,2 _ j („^y)2 . «^2. 210. We now come to the fundamental theorem in the present calculus, namely, that every invariant form can be re- presented symbolically by three types of factors, viz. (a/37), («/S'0 ^-J^d Oy. together with the identical invariant form iix. * 209-211] TERNARY FORMS 259 Some preliminary observations will enable us to simplify the proof. In fact an invariant form containing the u'» may be regarded as a pure covariant of the original form or forms together with the linear form and hence if the theorem be proved for any number of ground forms for concomitants in which the us are absent the general case follows at once by adjoining a linear form to the original system. Let us then proceed to prove the theorem for invariants and covariants. In the proof we denoted by need two lemmas regarding the operator O d ail d d a d a dvi dv. dVs d ari d d ara which is the natural extension to three variables of the operator so important in the binary theory. 211. I. // D denote the determinant li I. la Vi V-2 Vs ?I ^. ^3 then H*'!)"' is a numerical multiple of D"-*", where n, r are positive integers and r ^ n. This result can be proved for r = 1 by straightforward differentiation and then the general result follows. We may shorten the proof by using properties of the minors of the determinant. For a moment denote the minor of |i by Xj, that of 7}^ by Fj, -of ^1 by Z^, etc. 17—2 260 THE ALGEBRA OF INVARIANTS [CH. XII Then and p^ = n (n - 1) D'-^ F3Z, - nD"-' f ^ . 0973d^2 Hence B'^^iiin-l) D"^- ( Y.Z, - Y,Z,) + 2nZ)"-i |, and Y,Z,-Y,Z,= ^,D by the properties of minors, hence / d^ d^ \ _ Consequently 4 (^^ - ^) ^" = ^ (»i + 1) (^ - 1) I)"-^^i^a + n{n + 1) i)"-S and adding to this two like terms we have nD'' = n(n + l){n-l) D"-^ {^.X, + ^^X, + ^,X,) + 3w (n+ 1) i)"-i = n{n+l){n + 2)D''-\ since Operating with fl again we have n^D^ = (n - 1) n^ (w + 1)2 (n + 2) D^-^, and proceeding in this way we see that D>^B^ is a numerical multiple of D^~^ and, in particular, n**/)" is a numerical constant. The reader will find it instructive to extend this property of the operator H to four or more variables. 212. 11. If S be a product of m factors of the type a^, n factors of the type ^,, and p factors of the type 7^, then iVS is the sum of a number of terms each containing r factors of the type {oL^y), m — r factors of the type aj , n — r of the type yS, and p — r of the type 7^ . 211-213] TERNARY* FORMS 261 The proof of this result is exactly similar to that of the corresponding theorem in binary forms. In fact let S = PQR, where then ^'^ - = y a, w P (") yj^) — _^ -^ the sum being taken for /Li = l, 2, ... m; V = 1, 2, ... n ; TV =1,2, ...p. Writing down all six such terms and adding we have P which proves the theorem for r = 1, since — -- contains m — 1 factors, and so on. Now £1 has no effect on a factor (a^y), hence applying the same result to each term in H . >S we obtain the result for r = 2 and so on for any value of r. In particular if r = m = n=p the result is expressed exclusively in terms of factors of the type (a^7). 213. Our proof that any invariant or covariant can be repre- sented by factors of the two types ((X0y), a^ follows exactly the lines of the second proof of the corresponding theorem for binary forms. As the proof is exactly the same for any number of forms, we give it explicitly for one only. First suppose that / (a) is an invariant of the form nl -^^ p ! g- ! r ! then if ^Ay^p. — ^ — j — , X-^X.^X^ be the transformed expression, we have /(^) = /(a).(|i70"' identically. 262 THE ALGEBRA' OF INVARIANTS [CH. XII Now express the left-hand side of this equation in symbols. We have cipqr = cL^oi2^a/, and hence ^^^ = ixi^a^^a^ *■. Thus on the left we have the sum of a number of terms. Each contains w factors of the type a^, w of the type a,, and w of the type a^. Hence operating on both sides of the equation with H^, the left- hand side becomes an aggregate of terms each made up entirely of factors of the type (a^y) and the right-hand side is a numerical multiple of 1(a), so that we have expressed I (a) in the required manner. The proof for a covariant is similar in form to the above, but as in the case of binary forms, a little care is necessary. Of course, here as always, we confine ourselves to co variants that are homogeneous in the variables x^, x^, x^. Suppose that G {a, x) is a covariant of order m, then by definition C{A,X) = {^ri^YxG{a,x). Now on solving the equations of transformation a?i = ^1^1 + Vi^^ + ^i-X'3, etc. we have ^' (M) ' On replacing Xy, by (^2*^3 - '^3^2), ^^2 by {v^Wi-v^w^) and x^ by iy^Wz — v^Wi) — a set of equations which may be written x = {vw) — we have X,= ^ ^V^Wf-V^W^ W( — V( Again, Apqr = a-^°^^o.i, and hence on substituting for the A'b and ar's their values as given above in G {A, X), and multiplying across by (^0"*> ^^e left-hand side becomes an aggregate of products of factors having the suffixes f , ?;, ^, and the right-hand side becomes (^t;^)'*^'" x G{a, x). Moreover each term 213] TERNARY FORMS 263 on the left-hand side must contain (w + m) factors with each siiflSx. Now if we operate on the left-hand side once with CI, we obtain an aggregate of terms each containing one determinantal factor and (tu + ni — l) factors with each suffix. The determinantal factors are of three types : (i) («yS7), (ii) (avw) = a^, (iii) (o^/3v), where the first two are of the form we require in the result, but the third is not. Now a term involving a factor (a^v) must have arisen from operating with 12 on a term containing a factor such as ol^^^v^ and this term must therefore contain a further factor w^ (or w^). Accordingly let the term be in the original expression then from the mode in which v and w occur in the expression on the left-hand side there must also be present a term and on operating with H this gives whereas the former term is But by the fundamental identity (a^v) Wr, + i^wv) a, + (wuv) 13^ = (wol/3) Vr, , or (a^v) Wr, - (a^w) v, = (fivw) a, - (avw) /3^ = /3x^r, - c^^r, , and the sum of the two terms mentioned is G{^xOL,-Oix^,). Thus although factors like {cl^v) do appear explicitly after operating with H, the terms containing them may be paired in such a way that the aggregate may be expressed entirely in terms of factors of the types (a/Sy) and cix together with factors having the suffixes I, 7], or ^. 264 THE ALGEBRA OF INVARIANTS [CH. XII Hence if we operate with H, lu + m times on the left and transform at each step as indicated above, we finally have an aggre- gate of terms each containing {w + m) factors of the types (a/Sy) and Ox only; also after performing the same operations on the right-hand side, we are left with a numerical multiple of C (a, x). We have therefore expressed the covariant as an aggregate of terms each of which is the product of a number of factors of the types (a/S7) and a-g, and since the order must be m, it follows that in each term there are lu factors of the type (a/87) and m of the type a-B. 214. Leading Coefficients of Concomitants. In § 33 it was shewn that a covariant of a binary form could be deduced from its leading coefficient; we shall in what follows consider the extension of this idea to ternary forms, but as the results are easily obtained and not necessary for present purposes our remarks will be somewhat brief ; the reader who desires further information is referred to a memoir by Forsyth, Am. Journal of Mathematics, 1889. If a mixed concomitant be of class m' and order n', the coefficient of x-^'u{^' is called the leading coefficient. Given the concomitant, the leading coefficient is unique of course, but the reverse is not true, because if we multiply a concomitant by any power of Ux we obtain another concomitant with the same leading coefficient. However, save as to a power oi Ux, a concomitant can be found when its leading coefficient is given, as we proceed to shew. The leading coefficient is an aggregate of products of factors of the types and on replacing a, by a^, oj by aj,, «3 hy oj, and so on, the above factors become {a^y)Ux, (a^u), Ox respectively, where u={yz). Cf. § 209. The symbolical substitution is equivalent to replacing the coefficient C'pqr by fi(^ so that if in a leading coefficient we replace Op,,. by the value just given we obtain the corresponding concomitant multiplied by a power of Ux — the ex- ponent of Ux is in fact equal to the number of factors of the type {afiy) that occur in the symbolical expression of the concomitant. 213-215] TERNARY FORMS 265 Hence, as in binary forms, it follows that " If there be an identical relation between a number of leading coefficients, the same relation exists among the corresponding concomitants." We leave the reader to establish this on the lines of § 33 ; the con- comitants corresponding to the various leading coefficients must be chosen so as to make the whole expression of uniform order and class. On multiplication by a sufficiently high power of Ux a concomitant can be completely expressed by means of factors of the types (a^«) and a*. The leading coefficient can therefore be expressed in terms of factors of the types ("2^3-03^2) and oi. It follows immediately that the leading coefficient is an invariant of the binary forms (02.^2 + 03^3)", (02^72 + 03.^3)" -ifli, {a,^X^_^->r a^X^ a^-\ which are n in number. These binary forms are the coefficients of the various powers of x^ in the original ternary form, hence they can be obtained at once from /. This is the fundamental result of Forsyth on algebraically complete sets of ternariants and from it readily flow all the results in the memoir cited. Similar methods may be applied to obtain the results of another paper of Forsyth's*. 215. We shall now explain an important principle due to Clebsch which establishes a connection between the invariants of binary forms and the contravariants of ternary forms. To facilitate the discussion we shall commence with a special case. Suppose we require to find the condition that the line Ur^ — should touch the conic a^- = h^- = 0. Let the points on the line be yxViVz ^■iid z-^z-^z^, so that Wi : u^ : M3 = ijoZ^ - y^Zo : y^z^ - y^z^ : y^z. - y^z^. The coordinates x^, x^, x^ of any point on this line are of the form Il2/l + l2^1> |l2/2+fe, I12/3 + 12-^3, and fi, ^2 may clearly be regarded as the coordinates of a variable point on the line referred to the points y and z as base points. We have * See London Math. Soc. Proc, 1898. 266 THE ALGEBRA OF INVARIANTS [CH. XII and the points in which the line meets the conic are given by (aj/fi + a,^,y = {hy^, + hU = . . . = 0. As the line touches the conic this expression regarded as a quadratic a^^ = ^^^= ... in ^ must have its invariant (a/3)- zero. Hence {aybz — azbyf = 0, and by an identity already given {Oyhz — agby) = (abu) ; hence we have {abuf = as the required condition. Ex. If tike line ttg = cuts the conies ax^ = 0, h:^=0 in harmonic points, prove that {ahuf=0. Thus when the points on the line satisfy the invariant relation {cL^y = 0, the line satisfies the relation {ahuf = 0. This principle is true in general, that is to say, in order to pass from the invariant relation satisfied by the points in which the line meets any number of curves to the condition satisfied by the coordinates of the line, we have merely to change every factor of the type (a/3), in the expression of the binary invariant, into a factor {ahu). The proof is very simple — suppose the equations of the curves are a^'« = 0, K'' = 0, etc., and that y, z are two points on the line, then the points in which the line meets the curves are given by the binary forms (ayfi + a^^,)^ = 0, {hy^, + h^,Y = 0, etc. But an invariant of these forms is expressible in terms of factors of the type {ayh^ - azby) entirely, and inasmuch as a-ybz - ttzby = (ahu) the result follows at once. Thus, knowing the discriminant of a binary form of order n, we can find the tangential equation of a curve of the nth degree, for the discriminant being equated to zero gives the condition that 215-217] TERNARY FORMS 267 two points of ntersection should coincide, and in this case the line touches the curve ; e.g. the discriminant of a binary cubic being {ahf (ac) (bd) (cdf, the tangential equation of the curve dx^ = hj = Ca;^ = dx^={) is {ahuf (acu) (bdu) {cduf = 0. 216. The principle can be extended to covariants and mixed concomitants, as we shall explain by an example. A line Ma; = cuts two conies a^^ = 0, bx^ = in pairs of points PQ, RS: to find the pair of points harmonic to both P, Q and R, S. We know that the pair of points harmonic to those given by af2 = 0, 0f = 0, are given by equating the Jacobian (a/S) a^ yS^ to zero. Now if 2i be the join of the points y and z, then fi, ^2 are given by (a,^i + ch^.Y = 0, {by^, + bSy = 0, hence the common harmonic pair are given by (ayb^ - Uzby) (ay^i + a^^a) (by^i + h^^) = 0, or they satisfy (abu) Uxbx = 0, and are therefore the points in which this conic meets the given line. The extension to covariants in general will now be sufficiently obvious. 217. There are also dual methods which enable us to deter- mine, for example, the locus of a point such that the pencils of tangents drawn from it to a number of fixed curves possess a given projective property. After the study of a simple example it will be easy for the reader to enunciate and prove the necessary theorems. Let us find the locus of a point such that the pairs of tangents drawn from it to two conies Ua' = 0, u^^ = 0, are harmonically conjugate. 268 THE ALGEBRA OF INVARIANTS [CH. XII If any two lines through the point are v^ =0, Wx = 0, then the coordinates of a variable line through it are where |i, ^2 may be regarded as the coordinates of a variable line in the pencil. The tangents to the two conies are given by the binary quadratics (vaai+u^a-j + Usa^y, or (Wa|i + Wa|2)' = 0, and (vp^i + 'i^p^2y = 0. And since these are harmonic {VaWp — V^Way=0, bu t VaWfi — V^ Wa = {OL^X), where x is the point of intersection of the lines v and w. Hence the required locus is {a^xf = 0. . If a and y8 are equivalent symbols, this gives the point equation of the conic whose tangential equation is W„2 = w^2 = 0. Ex. (i). Prove that if three binary quadratics a^^, h^, c^ are in involution, then {ab) (be) (ca) = ; deduce the locus of a point such that the tangents from it to three conies form a pencil in involution, and state the correlative theorem. Ex. (ii). Interpret the envelope loci {abu)* = 0, {abuf'{hcuf{caiCf = in connection with the quartic curve ax* = bx* = Cx* = 0; shew that the inflexional tangents touch each of the above curves, and are therefore in general com- pletely determined as the common tangents. Ex. (iii). A straight line p meets a cubic curve in P, Q, R, shew that (i) there are two points IT, W on the line such that their polar conies touch the line ; (ii) the polar conic of H touches the line in H' and that of E' touches the line in H ; (iii) H and H' are the Hessian points of P, Q, R; (iv) if p pass through a fixed point y the locus of H and H' is the quartic obtained by putting u={xy) in {abuy axhx'^0 ; (v) the equation of the quartic is fP-C'^ = 0, where /=0 is the cubic, C the first and P the second polar of y. 217-218] TERNARY FORMS 269 218. Quaternary Forms. The general quaternary form of order n is plqlrl si where , p + q + r + s = n. In accordance with the methods used in this work, we write this in the umbral form and this in turn we denote by ax\ To represent expressions of degree higher than unity in the coefficients, we have to introduce equivalent symbols yS, y, etc., so that /=a/^ = /3^" = 7^'' = etc. Denoting the determinant «! a., O-i 0-i ^1 /3. /33 /34 7i 72 73 74 B^ 8, K ^4 by (a/SyS) and extending the definitions of invariants and co- variants, we can shew that every expression completely represented by factors of the type (a^jB) is an invariant, and every expression represented by factors of the types (a^yS) and ax is a covariant ; of course in a symbolical expression each symbol must occur to degree n. Then using plane coordinates u, such that UiXi + U2X2 + U3X3 + iiiX^ = is the equation of the plane ii, we have mixed concomitants containing the three types of factors ax, (a^78), (a^yu). Finally, introducing a second plane v, we have a more compre- hensive type of mixed concomitant expressed by factors of the four types ax, (a^7v), (a^yu), (a^uv). There is no need to introduce a third set of plane coordinates w, because {auvw) / 270 THE ALGEBRA OF IXVARIANTS [CH. XII is really of the form ax, where x is the point of intersection of the planes u, v, lu. The reader will not have more than purely algebraical diffi- culties in extending the methods of §§ 211, 212 to four variables. 219. In illustration of the foregoing we may mention that if / = a^' = ^x^ = li = ^x be a quaternary quadratic, then (i) {oL^^hy^ is its discriminant ; * (ii) {a^>yuf = is the condition that the plane u should touch the quadric /= ; (iii) {oL^uvf = is the condition that the line of intersection of the planes u and v should touch the quadric ; (iv) (auvwY = is the condition that the point of intersection of the three planes u, v, and w should be on the quadric. 220. It is clear that when any invariant form expresses a relation of a projective nature between a straight line in space and a surface, factors of the type (a^uv) must occur in its symbolical form, because the straight line is given in the first place as the intersection of two planes, which are u and v in this case. It is easy to see that (a^UV) = S (ttiySa - ftzySi) (1*3^4 - W4V3) ', the six quantities MjVg — W3V2, UsVi — UjV-i, KtV-^ — U^Vi, UiVi-tUVi, U^Vi — UtVi, UsVi-UiVs are called the six coordinates of the line of intersection of the planes u and v. It can be verified that they are altered in the same ratio, when instead of the planes u, v we take any other two planes through the line. Further, if cc, y be any two points on the line, the quantities • a;i3/4-^42/i. Xc^A-Xiy^, x^Vi-oc^z, ic^yz-x^y^, ^syi-^iya, ^12/2-^22/1 «1 Xn a?3 «4 2/1 2/2 2/3 2/4 Wl W2 M3 %4 Vi ^2 V3 ^4 218-221] TERNARY FORMS 271 are altered all in the same ratio, when instead of x, y we use two other points on the same line. Finally, from the equations UX = 0, Uy = 0, Vx= 0, Vy = 0, it follows that ^l2/4 -X^^ = p (U.V3 - U3V2) =pu, ^22/4 - a?4y2 = P (W3V1 - W1W3) =^24, .-^^32/4 - «42/3 = P ("1V2 - M2V1) = i>34, ^22/3 - ^32/2 = P (^^^4 - W4V1) = P23 , ^3^1 - ^l2/3 = /3 (^<2'y4 - ^4^2) =P31, ^l2/2 - ^22/1 = P (^^4 - W4V3) = ^12 , in other words, that the six coordinates of a line are either the six determinants of the array or those of the array The expression (a^iw) is written (a^p) for convenience, but it must not be confused with the similar expression for a three- rowed determinant. The discussion of concomitants involving line coordinates is complicated by the existence of the relation PtsPu + P31P24. + PiiPsi = between the six coordinates of a line. 221. As a simple illustration of the use of the above methods, we shall take the extension of the principle of Clebsch explained in § 215. To explain this extension we take a particular problem, viz. to find the condition that the plane Wa, = should touch the quadric surface represented by (^x- = yS^' = jx- = 0. Let X, y, z be three points in the plane, then any other point (Xi, X2, X3, X4) in the plane is given by Xr = f 1.^. + t^Jr + Ur, ^ = 1 , 2, 3, 4, 272 THE ALGEBRA OF INVARIANTS [CH. XII and ^1, ^2, ^3 may be regarded as the coordinates of (X^, X^yXs, X^) in the plane referred to x, y, z as fundamental triangle. 4 Now 'la.rXr = aa;f 1 + O^^s + fl^^s , 1 hence the equation of the conic in which the plane meets the quadric may be written If the plane touch the quadric, this conic must be a pair of straight lines ; the condition for this is O-x ««/ ftz ^x ^y ^z 1% ly Iz = 0, and inasmuch as the determinant here written is equal to the condition required is (a/37M)2 = 0. The reader will have no difficulty in applying the same method to find the envelope of a plane cutting a surface in a curve which has a definite projective property, of which the invariant equivalent is given. 222. We can in like manner solve problems leading to line coordinates, e.g. to find the condition that a line should cut the quadrics a^' = 0. ^9x^ = 0, in pairs of points harmonically conjugate. Let X, y be two points on the line, then any other point X is given by Xr = ^lOCr + ^23/r, ^ = 1, 2, 3, 4, and ax = cl^^i + Oy^i, so that the quadratics giving the ratio in which the line is divided by the quadrics are 221-222] TERNARY FORMS 273 The condition that the pairs should be harmonically conjugate is {^x^y - tty^xf = 0, or (a/3py = 0, where p is a typical coordinate of the line. The construction of other examples and the discovery of the dual principles will be easy for the reader who has grasped the corresponding results relating to ternary forms. G. & Y. 18 L CHAPTER XIII. TERNARY FORMS (continued). 223. In this chapter we shall give some further theorems relating to ternary forms; mainly such as arise in obtaining the irreducible systems in the simpler cases. In connection with Hilbert's proof of Gordan's theorem it has already been remarked that this proof can be extended without much difficulty to the case of ternary forms; we are therefore certain that for any number of such forms the irreducible system is finite, i.e. there exists a finite number of invariant forms in terms of which all others can be expressed as rational integral algebraic functions. But although the existence of the finite system is thus established, no clue is. given as to the method of discovering such systems and, as a matter of fact, very little is known in this branch of the subject. The more important complete systems known up to the present are those for one, two* and three f ternary quadratics, that for a single ternary cubicj form, and quite recently Gordan has given the complete system for two quaternary quadratics§. 224. In his paper on the ternary cubic Gordan gave a syste- matic method of searching for all irreducible forms by proceeding from those of degree r — 1 to those of degree r in the manner of Chapter v. ; in any but the simplest cases the application of this method requires uncommon skill and patience. The reader will find an introductory sketch of the method at the end of this chapter ; we shall content ourselves with obtaining the complete systems for one and two quadratics by an elegant and ingenious * Clebsch, Geometric, p. 174. t Ciamberlini, Battaglini, Vol. xxiv. X Gordan, Math. Ann., Band i. § Math. Ann., Band lvi. i 223-225] TERNARY FORMS 275 process, also due to Gordan, which has been applied with success to three, and would probably be equally successful in dealing with any number of, quadratics. The complete systems are in all cases, except for linear forms or one quadratic form, so large that any method for their discovery will involve a great deal of labour. Ex. (i). For a single linear form a^ the complete system is «„. Ex. (ii). For two linear forms a^ and b^ the complete system is a^, b^, {abu). Ex. (iii). For any number of linear forms a^, b^, c^, ... the complete system consists of (a) Forms of the type %, (/3) Contravariants of the type (abu), (y) Invariants of the type (aba). These results all follow immediately from the symbolical expressions for invariant forms. Cf. § 84. 225. One Quadratic Form. Let the form be f^ai = h^ = Co? = etc., then the t3rpical invariant factors are (abc), (abu), a^, and there are three invariant forms (abcf, {abuf, a^ ; we proceed to shew that these constitute the complete irreducible system. I. Any invariant form P containing the factor (abc) can be transformed so as to contain the factor (abcy. First suppose that no two of the letters a, b, c occur again in the same factor, then P = (abc) apbqCr M, where M does not contain a, b or c, and p = (de), (dib) or x etc. Hence by interchanging a, b, c in all possible ways 6P = (abc) M(apbqCr — apbrCq — aqbpCr + ttqbrCp — arbqCp + arbpCq) — (abc) M(abc) (pqr), = (abcf M (pqr). 18—2 276 THE ALGEBRA OF INVARIANTS [CH. XIII Thus P involves the factor {abc)- which is an invariant, hence the other factor must be an invariant form* and P is therefore reducible. Secondly, suppose that P = (abc) (abp) CqN, where p = d or u, q = (de), (du) or x. Then 3P = (abc) iV {(abp) Cq + (bcp) a^ + {cap) bq] = {abc) N {abc) pq = {abcyNpq, and as before it follows that P is reducible. Thus any form containing the factor {abc) is reducible, and we may neglect such forms in future. . II. A form Q containing the factor (abu) can be expressed in terms of forms containing the factor {abuf and reducible forms. Rejecting forms containing factors of the type {cde), we have the three following possibilities : (i) Q = {abu) {acu) {bdu) M, (ii) Q = {abu) {acu) b^ M, (iii) Q = {abu)aJ)xM. As regards (i), on interchanging a and b, 2Q = {abu) M [{acu) (bdu) + {cbu) {adu)] = {abu) M j — {bau) {cdu)] = {abuy {cdu) M, and Q is reducible. (ii) Here 2Q = {abu) M [{am) b^ - {bcu) a«} = {abu) M [{ahu) Cx — {abc) Ux] = {abuf Mcx — {abc) u^ {abu) M, and the latter part is reducible because it contains the factor {abc). Thus again Q is reducible. * We shall return to this point later, see § 226. 225-226] TERNARY FORMS 277 (iii) In this case Q vanishes, as we see by interchanging a and b. Now every symbolical expression representing an invariant form contains a factor of one of the types (abc) or (abu) unless it be ajT] it follows that all such are reducible except (abcy, (abuY, a^-, and these form the complete irreducible system. Of course, to be strictly accurate we should add to this and every complete system the identical concomitant u^. Ex. (i). Express (abu) (acu) b^c^ in terms of the irreducible system. Ex. (ii). Prove the symbolical identity {ab (pq)} = apb, - aj)p . 226. The foregoing discussion leads to some general remarks that will be useful in the sequel. The artifice of replacing {abc) by cip, or, what is the same thing, writing {bc)=p, and more explicitly (62C3 - 63C2) =Pi ; (&3C1 - 61C3) =^2 ; {biCi - b^Ci) = ps, will be frequently used. (It is clear that if b and c be two straight lines then p is their point of intersection.) In particular the contravariant {abuf will be written u„^, thus a = {ab\ and the invariant of the quadratic is Ca. The a's may be regarded as expressible in terms of the original symbols, but we shall more often regard them as independent symbols ; the contravariant above will thus be 2_ „,2_^ o, 2^ the original form is J ^ tt/g" ^ Of' ^ Cat" = . . . , and the invariant is «a" = iifi^ = b^- = etc. Of course in replacing an a by symbols belonging to /, we must use no symbol already occurring in the expression because 278 THE ALGEBRA OF INVARIANTS [CH. XIII then that symbol would occur more than twice in the transformed expression. The a's being formed from the a's and &'s, as line coordinates are formed from point coordinates, it follows that the a's are contragredient to a and u but cogi'edient with x. We shall now return to a point that arose in the discussion of a single quadratic, where we deduced from first principles that if Op, ttq, ar be all invariant factors then {^qf) is an invariant factor — it will be shewn here how to express it in terms of factors of the types {ahc), (abu) and a^. The possible forms for p, q, r are (de), (du) and cc ; if each of them were cc, (pqr) would vanish, so we may suppose that p for example is not x. Let p = (dv), where v is either e or u, then {pqr)={{dv)qr}=dqVr-drVq, (§ 209) but since bq is an invariant factor, dq and Uq are both invariant factors ; for if in such a factor we replace a letter of the type a by another of the same type the factor is still invariant, while if we replace it by it the resulting factor either vanishes or is an invariant factor. Hence (pqr) can be. expressed entirely in terms of factors of the types (abc), (abu) and a^ ; it is therefore an invariant factor. 227. To make the matter clearer we shall illustrate it by some examples. Suppose a quadratic is U/X — ^X — ^X — • • • > and its contravariant is then tta, ap,ba, bp etc. are invariant factors. Hence (a fix) must be an invariant factor and (a/Bxy an in- variant form of the quadratic. We proceed to express it in terms of the members of the irreducible system as follows : (afixy = (ab fixf = (apb^-ax bpf = a^^ . b^ + a^ . b^ - la^b^a^b^, and we have now to deal with the term Uxbxapbp. 226-227] TERNARY FORMS 279 As a guide to the further reduction we remark that since a^ is a factor the form must involve the factor a^^ after suitable trans- formation. Thus ajy^a^h^ = a^bx (acd) (bed) = (bed) ax (acd) bx, and Saxbxa^bfi = (bed) ax {(acd) bx — (abd) Cx — (acb) dx}, or since (acd) bx — (abd) Cx — (acb) dx = «« (bed), Sttx bx ap bp = (bcdy a^. Consequently (a^xf = 1a^ . bx' -^aj. bx' = ^a,' . bx', and except for a numerical factor the invariant form is equal to the product of the original form and its invariant. The following example will shew the advantage of introducing symbols like a, /8, 7 above in geometrical investigations. Let ax =0,bx = 0,Cx = be the equations of three straight lines, then if a = (be), /3 = (ca), 7 = (ab), (^y) = (abc) a. This is not surprising, because a, ^, 7 are the points of inter- section of b, c; c, a; a, b respectively and (^y) must therefore give the line joining ^ and 7, that is a*. The equation of the straight line joining the point y to the point of intersection of ax = and 6^ = is (ab, xi/) = or axby — aybx = 0, which is the well-known form. The line joining the point (a, b) to the point (c, d) is (ab, cd, x) = 0, or (acd) bx — (bed) ax = 0. This may also be written (abc) dx — (abd) Cx = 0, and the two forms are equivalent in virtue of the fundamental identity from which both can, in fact, be at once inferred. Now let a', b', d be the sides of a second triangle and a', ^ , y' its angular points. * The similarity of these ideas to the methods of Grassmana's^iisdfift/mjjgfs/e/ire will not escape the notice of the reader who is acquainted with the Calculus. 280 THE ALGEBRA OF INVARIANTS [CH. XIII The lines joining corresponding vertices of the two triangles are (aa), (yS/3'), (77'). and these are concurrent if i.e. if (a^/3') (a'77') - (a'/3/3') (077') = 0. But since (ayS) = (abc) c etc. this condition is (abc) {a'h'c') [c^'h'y-c'^hy,] = 0, or {abc) {a'b'c') [{a'c'c) {abb') - {ace') {a'b'b)} = 0, or {abc) {a'b'c') [{abb') {a'cc') - {ace') {a'bb')] = 0, or {abc) {a'b'c') (a^', W, ~^) = 0. Now {aa', bb', cc') = is the condition that the points of inter- section of corresponding sides should be collinear. Hence when the joins of corresponding vertices ate concurrent the intersections of corresponding sides are collinear — a well-known theorem. A direct analytical proof by the ordinary methods without using a particular triangle of reference is by no means easy. Ex, (i). Prove that the cross ratio of the range in which Ux=0 is met by ax=0, bx = 0, Cx=0, dx=0 is — • \\ ^ i i hence the general equation of a conic touching the four hues is (abu) (cdu) + X {bcu) {adu)=0. If the six hnes a, b, c, d, e, f touch a conic, then (abe) (ede) (bcf) (adf) = {bee) (ode) (abf) (cdf). Ex. (ii). State and prove the results dual to those of (i). Ex. (iii). If the points a, /3, y, 8, e, f be on a conic and {a^)=p, m = q, (yS) = r, (8e)=y, «) = ?', {Co) = r\ then {pp', qq', rr')=0. Deduce Pascal's theorem and prove Brianchon's theorem in the same way. 228. Two Quadratics. Suppose the quadratics are y = aJ' = bx^ = Cx^ = etc., /' = aV = 6V = cV = etc. and that their contravariants are (f) = uj = n^ = u.^ = etc. ^ = uj' = Ufi'^ = Wy2 = etc. respectively, then their invariants are aa^ and a ^^ respectively. 227-228] TERNARY FORMS 281 The types of invariant factors are (abc) = tta, (aba) = a„, (ab'c) = «„', (a'b'c) = a'af, (abu) = »„, (a'b'u) = Ua', (aa'u), Ux, a'x, and in the course of the work we shall use the additional types {a^x), (a'/3'a)), (aa'x) which, as has been pointed out, certainly are invariant factors. There are four types of symbols occurring in any expression, viz. a, a', a, a', and the leading idea of the investigation is to reduce the number of symbols in the expression for any invariant form to a minimum. Thus {abuy involves two symbols a, b, but it can be written Ua^ and then only involves one. Again 2a„,'b„.'axbx is equivalent to {aa'bx — cixb„,)^ except for reducible terms ; then since aa'bx— Oxbo: = (oLOL x) we see that aa.'ba,'axbx can be expressed in terms of reducible forms and forms containing a smaller number of symbols. This example will make clear what we mean in the sequel by reducing the number of symbols. I. // a symbolical product contain a factor of the type a^ or a' a' it is reducible. For let P = (abc) ttphqCrM, then, as in § 225, eP = (abcy.(pqr)M, and ( pqr) is an invariant factor ; hence P is reducible. This shews incidentally how factors of the type (a^x) arise naturally in the course of the work, for we might have p = a, q = ^, r = x. 282 THE ALGEBRA OF INVARIANTS [CH. XIII The case in which a and h occur again in the same factor, e.g. (abu), is treated as in § 225. II. If a symbolical product contain a factor of the type {ahv), where v is of the type a' or is u, then it can be so transformed as to contain an additional factor (abw). Suppose in fact that F = {abv)apbqM, then 2P = (abv) (apbq — aqbp) M = (abv) {ah, pq) M = Va. (otpq) M, and (apq) can be expressed in terms of a and the symbols occurring in p and q. Hence in the transformed expression there are two factors involving the symbol a. A like theorem applies to a', and here we see a great advantage in introducing these symbols, because just as when one a occurs in a product there must be another, so when a occurs the expression can be transformed so as to be of degree two in a. III. If an expression contain a factor of the type (a^y) or {a'^'y) it is reducible. For let P = (a^y) M, where y must be of the type a, a' or x. Then since (a^y) = (ab, ^y) = a^by-b^ay, P = a^byM-bfiayM and by I. the form P contains the invariant a^- as a factor. Summing up these results we infer that any concomitant other than a^^, aV, ctaS a'a^ must be composed of factors of the types ««. a'sc. c^a', a'o, Ua, Ua', {olol'x), (aa'u), and if one a occurs there must be another present after a suitable transformation. 228] TERNARY FORMS 283 IV. Any expression containing two equivalent symbols can he expressed in terms of concomitants that are either reducible or contain a smaller number of symbols. There are two cases according as the equivalent symbols are of the type a or a, so that we have to consider the expressions apaqbrbgM and ■sr^paa'pTpM, with the exactly similar ones a'pa'qb'rb'gM and ■uTa'paO-p'r^'M. Since a^ br — ar bp = (apr), and ttg bg — agbq = (aqs), it follows that expressions derived from apaqbrbgM, by permuting the symbols p, q, r, s, differ from the original ex- pression by forms in which the two symbols a and b are replaced by the simple symbol a. Hence if any one of the three expressions apaqbrbgM, aparbqbgM, apagbqbrM can be expressed in terms of reducible forms and forms containing fewer symbols the same is true of the other two. The same result can be easily established for the three expressions -^aPaO-^T^M, 'UTa.ar^p^T^M, 't^a.Ta P^CT ^M ] in fact the difference between any two is reducible by III. Consider now the expression apaqbrbgM. There are only five possible types for p, q, r, s, viz. a, a, (bu), (a'u) and X ; of these a may be neglected because if it occur the form is reducible, and (bu) may be neglected because if it occur we can replace a and b at once by the single symbol a. Hence there are only three remaining possibilities, viz. a', (a'u), X, and oi p, q, r, s some two must certainly be of the same type. First suppose that two are identical, say p and r, then apUrbqbgM^ a.^ . bqbgM ; 284 THE ALGEBRA OF INVARIANTS [CH. XIII thus aparhqbgM is reducible, and hence cipaqbrbsM can be expressed in terms of reducible forms and forms containing fewer symbols. Similar reasoning applies to the expression when any two of the symbols w, p, a, r are identical ; here, as in the other case, there are three possibilities, viz. a, a' and u, so some two must be of the same type. Proceed now to the general case in which no two of the letters p, q, r, s are identical. In this case some two must be equivalent without being identical. Suppose the equivalent symbols be p and r, then there are three cases to consider. (i) Let p = r = x. Here p and r are identical and the result has been established already. (ii) Let p — OL, T — fi\ then ftp ar bq bg M = tta' cip' bq bg M. And since the symbols a, ^ each occur once the expression can be transformed by II. so that each occurs twice, hence aparbqbgM= a^' a^' ja- 8^' ^. But aa-ap'ja'B^'N falls under that class of expression 'STa'Pa.XTp'Tp'M in which two of the symbols bt, p, a; t are identical, for -07 = a- = a ; hence, as we have seen, it can be expressed in terms of simpler forms. Thus QpttrbqbgM can be expressed in terms of forms that are either reducible or contain fewer symbols, and the same is true of apaqbrbgM. (iii) Let p = (a'u), r = (b'u), then ttpQrbqbgM = (aa'u) (ab'u) bqbgM. And the latter expression can be written {a'au) (b'au) a't^b'^M', or a'p'b'ya'q'b'g'M', where p = (au), r' = (au), i.e. p' and r' are identical. 228] TERNARY FORMS 285 Thus, as we have ah-eady shewn a'p'a'q'h'r'h'^M' can be expressed in terms of simpler forms ; hence apUrbqlgM can be expressed in terms of forms that are either reducible or contain fewer symbols, and the same is true of apaqbrbgM. This completely establishes the Theorem IV. when the equi- valent symbols are of the type a or a. We have still to consider the general expression where -ot, p, a, r are each of the types a', (ax) or u. The case in which some two are identical has already been discussed and reasoning exactly similar to that of (i), (ii), (iii) above enables us to prove Theorem IV. for the general expression. The theorem we have just proved simplifies vastly the evolution of the irreducible system inasmuch as it shews that in an irreducible form there cannot be more than one symbol of any of the types a, a', a, a.'. A further limitation is imposed by the following theorem. V. A form containing hoth the factors (aa'u) and (aa' cc) may be rejected in constructing the irreducible system. In fact by direct multiplication (aa'u) X {aoLx) = I a„ a'^ Ua a^f a a' Ug^ I ax a X Ux hence if a form involve both these factors it can be transformed so as to only contain the simpler factors tta, a' a', a' a, ^a', Ua, Ha', ««. d'T ^nd Ux- With the aid of the above five theorems the problem of the complete system is reduced to a very simple one ; in fact we have only to write down such products of factors of the types tta, a'o', da', a a, Ua, Ma', ««;, Cl'x, (tta'u), {aCLX), as satisfy the conditions implied in the theorems. 286 THE ALGEBRA OF INVARIANTS [CH. XHI Since no two equivalent symbols can occur in the same product we need introduce no more symbols ; further a^, a'a- can only occur in the invariants aa^ and a'a'^ respectively ; therefore it only remains to write down all products of the factors da', of a, Ua, Ua', tt^, tt'a;, (tta'u), (au'w), in which every letter except u and x which appears at all appears twice and no more ; besides the two last factors must not appear in the same product. Hence we have the following forms : , (A) aa\ aV, «»'', a'a^ u^, w.-^, a^, aV, {av^uf, {(idxf, obtained by squaring each factor. (B) aa'ttajWa', aa'ax{oLOLx)Ua., aa'ax(aax)a'aa'a;, ac,'{aa'u)a'x, aa.'{aa'u)a'aUaUa', containing the factor a^'. (C) a'aa'xUc,, a'aa'x{o.OLx)ua', a'a{aa'u)ax, containing a a but not tta-. (D) Ua{a.OLX)Ua,>, containing Ua but neither a\ nor a^'. (E) ax{aa'u)a'x, containing a^ but neither a'^ nor a^'. There are twenty forms in all, viz.: Four Invariants, Four Covariants, ax^ = Si, a'x^ = 8^, {oLOLxy = F, {ao^x)ao:a\axa'x. Four Contravariants, tij = 1i, Ua'^ = l2> {aa'uf = ^, (aau)aa'a'aUaUa'. Eight Mixed Concomitants, cixda'Ua', a'xa\Ua, {aa'u)axa'x, {aa'u) aa-a xU^.; {aa'u)aaaxUa, (aa'x)Uaaa'ax, (aa! x)Ua' a' aa'x, {aOLX) U^Ua.: To these twenty should be added the identical concomitant %. 228-229] TERNARY FORMS 287 It should be noticed in conclusion that, although it follows from the five theorems of this article that every other invariant form is expressible in terms of these as a rational integral algebraic function, it has not been shewn that these twenty forms are themselves irreducible. Cf note, p. 131. Ex. (i). Prove that Ex. (ii). Prove that and thence that (««', aa')2 = |^Jjjj4222 + -^n2-^i22' Ex. (iii). Prove that Ex. (iv). Prove that a„u.a^(aa'x)=0. a a X ^ / 229. A complete irreducible system of invariant forms may be regarded as giving rise to two inquiries. (i) What is the geometrical meaning of each member of the irreducible system ? (ii) What is the expression in terms of the members of that system of the invariant forms which arises in the analytical treatment of a given problem ? To the first of these inquiries an answer can generally be given, provided a sufficiently complex geometrical apparatus be allowed, but it commonly happens that the significance of some members of the system is so remote as to render them of little geometrical importance. The second inquiry is naturally unanswerable until the pro- blem be named, and thus all we can do is to illustrate it by the discussion of some simple problems. Before going further it may be well to add that the second inquiry is the really important one ; in a manner it includes the first as a particular case, and in fact there being no direct method of proceeding from the invariant to the geometrical meaning, the answer of the first inquiry is obtained fortuitously in pursuing the second. If it be not obtained we should console ourselves with the reflexion that the uninterpreted forms are of little geo- metrical interest in the present state of knowledge ; besides, if we 288 THE ALGEBRA OF INVARIANTS [CH XIII regard the algebra as being merely helpful to geometry in the analytical formulation of results, it does not follow that everything in the algebra need be taken seriously from the geometrical point of view*. In spite of what we have said, we shall begin the geometrical theory of two conies by interpreting the members of the irre- ducible system ; it will be seen that they are all of importance in elementary geometry. 230. Geometrical Theory of tw^o Quadratics. The irreducible system. The forms themselves when equated to zero represent two conies, viz. a«=' = 0, aV = 0, which we write Si = 0, ^2 = 0, and call Si and S2. Invariants. The meaning of the invariant of a single conic is expressed by the fact that a J = is the condition for two straight lines. Again, aV = is satisfied when the point equation of Si involves only product terms and the line equation S^ only squared terms, or when the point equation of Si involves only squared terms, and the line equation of S.2 only product terms. It is then the poristic condition f that there should be an infinite number of triangles inscribed in the first conic and self- conjugate in the second, or, what is the same thing, that there should be an infinite number of triangles self-conjugate to the first conic and inscribed in the second. We shall consider this type of invariant more fully in the next chapter. Covariants, etc. The simple ones are well known, viz. ttx^ = is the condition that (x) should be on the first conic, Ua' = is the condition that (u) should touch the first conic. * Reducibility itself is a purely algebraical idea and the reader will soon convince himself that it is generally hard to obtain any geometrical satisfaction from the fact that a covariant is reducible. See a curious remark of Clifford's, Collected Papers, p. 81. t The sufficiency of the condition can easily be seen by taking a triangle inscribed in the first conic and having two pairs of vertices conjugate with respect to the second for fundamental triangle. 229-230] TERNARY FORMS 289 Of those not involving determinantal factors, there remain It will be sufficient to consider one of these. Now axofaU^ =■ is the polar of the point {a^u^, cuug,, a^Ua) with respect to aV = 0. This point is the pole of u with respect to the first conic u^- = or a^ = and hence a'xd'aiia = 0, when u is constant, represents the polar with respect to the second conic of the pole of u with respect to the first. When x is constant, it is the tangential equation of the pole with respect to the first conic of the polar of x with respect to the second. Again {aa'uf = is the equation of the envelope of the lines cutting the two conies in harmonic point pairs, and in like manner (aa'a;)"^ is the locus of a point such that the tangents drawn to the two conies from it are harmonic line pairs. Consider now the forms involving one determinant factor. (i) {aa'u) a^a X = is the condition that the lines a^ciy = 0, a sa!y = 0, iiy = 0, should be concurrent, y being the current coordinate, i.e. when u is constant it is the locus of points whose polars intersect on the line u, and when x is constant it is the equation of the point of intersection of the polars of x with respect to the conies. (ii) (aa'ic) iiaiw is dual to the last ; when x is constant it represents the envelope of a line such that the line joining its poles passes through x, and when u is constant it represents the equation of the line joining the poles of u with respect to the two conies. (iii) (aa'u) a'aa^Ua is the Jacobian with respect to y of the quantities Equated to zero, these represent three straight lines, namely, the polar of x with respect to the first conic, the polar with respect to the second conic of the pole of u with respect to the first, and the line u. The vanishing of the concomitant is the condition that the three lines should be concurrent; hence, for G. & Y. 19 290 THE ALGEBRA OF INVARIANTS [CH. XIII example, when u is constant, the equation represents a straight line constructed as follows : let Pj be the pole of u with respect to the first conic, and v the polar of Pj with respect to the second conic, then the line represented is the polar of the point {u, v) with respect to the first conic. (iv) (aa'a?) a^-a^a. is the Jacobian with respect to the i;'s of and hence equated to zero is the condition that these three points should be collinear. Now ujo^ = represents the pole of ii with respect to the first conic, and aafXaVa.' = represents the pole with respect to the second conic of the polar of x with respect to the first. Thus we can interpret the concomitant geometrically. See also Ex. ix. p. 294. (v) {aa'u) aa'a'aUaUa.' is a contra variant of the third class, and moreover the only such contravariant that can be built up from the members of the system. But the angular points of the common self-conjugate triangle must be given by equating to zero a contravariant of the third class. Hence the contravariant in question represents the vertices of the common self-conjugate triangle of the conies ax'^ = and aV = 0. We may prove this as follows * : The three conies ti^^ = 0, Ua'^ = 0, and {aa'uy = have a common self-conjugate triangle if there be a proper triangle self-conjugate to the first two, as we see by taking it for triangle of reference. Further, when three conies are referred to their common self-conjugate triangle, the Jacobian of their tangential equations is the tangential equation of the vertices. Hence the Jacobian of Wa^ Wa'^ {aa'uf represents the vertices of the common self-conjugate triangle. Now it is {aa'u) u„Ua' (aa'aa) — {aa'u) u^Uo.- (aaa'a' — aa-a'a). * But {aa'u) UaUa-aaa'a.' = 0, for otherwise it would involve the factors aa^ and aVS whereas its total degree is only six, so that the remaining factor would be a contravariant of zero degree which is impossible. * See also Ex. vi. p. 293. 230-231] TERNARY FORMS 291 Hence the Jacobian in question equated to zero is equiva- lent to {aa'u) UaUa'Cia-a'a = 0. (vi) By exactly similar methods we can shew that {aoix) aa.'Cia.a'xax = represents the sides of the common self-conjugate triangle. 231. Let us now consider some problems bearing on two conies with a view to illustrating the second inquiry of § 229. (i) To find the equation of the reciprocal of the conic So with respect to 8^. If y be a point on the reciprocal conic its polar with respect to €1x^ = must touch Uaj'= 0. The polar is a^ay = hxhy = 0, and if Ux = touches u^^ = / ^'^ ^2 ^2 \ '^ \dvidx^ dv-idxc, dv-idx^l ^ '^ °- >' Hence in this case d' d' d^ \% so that the reciprocal is axbxaa'ba' = or - (axK- - a^'hY + ax^ha,'^ + b:c-cta.''^ = or = la^'hoT- - {ololx)- = '±A^^. . S^ - F, which expresses the equation in terms of the irreducible system. (ii) To find the point equation of the covariant conic {aa'iif — 0. The point equation of u^ = is {a^x)- = 0, and hence in our case the point equation is [(m'hh'xY = Q, or {{ahh') a'^ - (a'bb') a^Y = 0, • i.e. (abby a'x' + {a'hh'f ax- — 2 {ahb') {abb') a^a'x = 0. 19—2 292 THE ALGEBRA OF INVARIANTS [CH. XIII Now (abb') (abb') a^a'x = — (abb') (a'ab') a'J)x = \(abb')a'x[aa'bx — ba-ax\, where a' = (a'b') = ^ (aoLx) (abb') a'^ = - i (aa'a;) (aba') b'x = -i (aa'a;) {a'^b'^ - a'^b'a] = i (aa'xf, thus the point equation is (abby aV + (a'bb'y a^ - \ (txrixf = 0, or J-na-'Sg + '^m'Sfi — i^=0. (iii) To find the locus of the point of intersection of harmonic pairs of tangents to F and >S. The locus for u^ =■ and iw^ = is (aaxf = and in our case the locus is accordingly [aa'axY = or (aa;a'a — a'jgaa)^ = 0, i.e. /Sfi^iia + 'Sa-^m — 2aa;a'a;aaa'a = 0. To reduce the last term we perceive that it must contain J-m and we write it (abc) a'x (a'bc) «« = i (abc) a'x [(a'bc) ax + (caa') bx — (aa'b) Cx] = ^ (abc) a'x (bca) a'x = i A^i . S^. The equation required is ^1-^112 + ^5^2-^ 111 ~ f 'Sia^iii = 0, or Aiu82 + SSiAu2 = 0. The following gives an easy means of verifying the results of (i), (ii), (iii) above and of the examples which follow. Taking the conies in the canonical forms Si =zaiXi^ + a^xi + a^xi O2 =^ X-i + x<^ + x^ we have Xi = 2 (a^a^vi^ + a^a-iui + a^a^u^) S2 = 2 (Ui^ + «2^ + ih% ^iu = 6a,a2a3, ^222 = 6, ^uj = 2 (ajas + as^i + Oiaj), ^122 = 2 (oi + aa + ftj), ^ = (ttj + rts) «i^ + (^3 + aO ui + (ai + Oa) 2*3^ jP = 4 {tti (Oa + ag) «i'* + as (03 + Oj) aja^ + 03 (a^ + aj) ajg^j . 231] TERNARY FORMS 293 where Q = Ex. (i). The locus of points whose polars with respect to S2 cut S^, S.2 in pairs of points harmonically conjugate is {aa'b') {aa'c') b'xc'x = 0, or A.222^i-SAi22>%=0. Ex. (ii). Prove that {abuy=^Q^ax^byhi,^) _s_ _3_ _a_ dxi dx2 8^3 d_ d_ d_ ^yi 8^2 ^Vz AAA 92, 3^2 823 and hence that (a^, a'/i', uY=^{<-A^^^A2a^{aa'iCf. Interpret this result geometrically. Ex. (iii). The line equation of F=0 is (aa! ^ m)2=0, or {a^^f uj- + (a'^^')' «.' " H^ ^' «)'• Thence (^' ^' w)2 = A{3^iii.4i2.^22+ 3^222-^11221 -2.4 111^222*}- Cf. Ex. (ii). Ex. (iv). Deduce from Ex. (iii) that the discriminant of F is {ad ^^ ryy = il (9^112^122 - -^111^222) ^111 -^222- Ex. (v). Deduce from Ex. (ii), (aa' 66' cc')^ =^(9^ j 12 ^122 --4 111^222) • Ex. (vi). Prove that the discriminant of {aa'u) a^a'^ is (aa'u) {bb'u) (cc'u) {ab'c') (a'bc). Hence if the conic (aa'u) axa'x = be two straight lines (aa'u) a ,a' u u ,—0. \ ' a a a a Thence verify that this equation represents the angular points of the common self-conjugate triangle and work out the dual results. Ex. (vii). Prove that if the point equation of a conic be Xi>S'i + X2'S'2 = 0, then its tangential equation is and its discriminant is Xi3J„j + 3Xi2X2.-lji2 + 3XiX22J,22 + V^222- Ex. (viii). Prove that the point equation of any covariant conic is of the form Xi,s'i+X2.y2+xi^=o, 294 THE ALGEBRA OF INVARIANTS [CH. XIII where Xj, X.,, X are invariants, and that its line equation is Xi22i + 2XiX2* + X2^22 + §XXi(Jiil22 + 3Jl222i)+|XX2(^2222i + 3Ju222) + X2A(3^jj^^j2222 + 3^222-^n22i-2Jin^222*) = 0- Ex. (ix). The locus of points whose polars with respect to S^ and F meet on the line u is (a aa u)ax{aa'x) = 0, or {aax)a^;u^a^ = 0. This gives a simpler interpretation of the irreducible form than that given in § 230. Ex. (x). Use the method of Ex. (ix) to interpret (aa'u)a'au^=0. ^ * a. X a. Examples (ix) and (x) enable us to interpret all the irreducible mixed con- comitants very simply in connection with the conies S^, /S'g, F, *. Ex. (xi). The locus of points whose polars with respect to F and * meet on u is ^112 {aax) a^u^a^, + ^122 {aa'x) a'^u^, a^,=0. (Use the point equation of * given in Ex. (ii).) Ex. (xii). The equation of the line joining the jjoles of u with respect to F and * is ^222-^112 (««'«) '^a%^'a + "^m ^^ 122 (««'«) «a'<*'a;«a' = ^^ (Use the result of Ex. (iii).) Ex. (xiii). Calculate the four invariants of the conies S^ and F. We have ^222 = 2^ (9^112^122 -^111^4222)^111^222 by Ex. (iv). (7,22 = (« ^' 0/3T = M3^'liii ^2^22 + ^222^112^111)- ^112 ==3 -^111 -^122 • ^111 = -^111- 232. Gordan's general method. Consider a concomitant of any number of forms containing the r letters a, b, c, ... h, L If we replace each factor of the type (aku) by a^, each factor of the type (abk) by {abu) and delete all factors kx the resulting expression is still an invariant form but only of degree r — 1 because it only contains r—1 symbols. Hence reversing the operation, i.e. replacing a proper number of factors of the type a^ by (aku), some of the type (abu) by (abk) and introducing a sufficient power of k^ we can deduce the form of degree r from one of degree r-1. Applying this process in all possible ways to all invariant forms of degree r - 1 we certainly obtain all invariant forms of degree r. We pause to explain more precisely what we mean by applying the reverse process in all possible ways to an invariant cf). Suppose in fact that the newly introduced letter k belongs to a form of order n, then we replace any p factors of the 231-233] TERNARY FORMS 295 type ax by iaku) any q factors of the type {ahiC) by {ahh) and multiply by k^ where of course we must have and we must take all values of jo, q, r satisfying this equation subject, of course, to the condition that there exist p factors of the given type to alter and a like restriction for q. For example, let the original form be {ahu){acu)hxCx, and suppose the new letter d like a, b, c belongs to a form of order 2. We can only change 2 factors at most in this case and we have p+q — l or 2, there are five cases, p = l, q = 0; p = 0, q=l', p = 2, q=0; p = l, q = l; p = 0, j = 2; and in following out the case 2) = l, q = l, r = 0, for example, we deduce four forms from the given one, viz. (abd) {acu) (bdu) c^ , {abd) {acu) b^ (cdu), (abu) (acd) {bdu) c^, {abu) {acd) b^ {cdu). The above indicates the general method of prcjcedure, but some introductory lemmas are necessary to render the method of any practical value — for example for all we know at present a form of degree r—\ which is identically zero might lead to irreducible forms of degree r, and we need hardly say that this would complicate matters enormously. 233. The reader will have observed some likeness between the above and the methods used in Chapter v. on binary forms to deduce the invariants of degree r from those of degree r- 1. This analogy will be fiurther exemplified in the rest of the argument. Consider the eflfect o^ the operator ii^' on the expression <f) being a covariant of degree r - 1 and 12 being the operator d d d 3a^i 3^2 dxs 3 a d 9^1 ^y'i 0^3 d d a dz^ dz. dz^ As we saw in the last chapter, the result is the sum of a number of terms each containing the determinantal factors of together with p of the form {akic), there being in the end p fewer factors of the type a^, {n-p) factors X-^ and no factors of the type u^ in each term. 296 THE ALGEBRA OF INVARIANTS [CH. XIII Next operate on the complete result of which a typical term is ■\^ky'^~P d^ 32 52 q times in succession. Since the effect of this operator on {abu)ky^~p is to give a multiple of (ahk)ky^~P~\ it is very easj' to see that the effect of the operator q times' is to give a number of terms each containing the same determinantal factors as ■<\r, with the exception that q factoi-s of the type {ahu) are replaced by {abk) ; the power of ky remaining in each term is ky^'^'t, and k replaces « in q places in all possible ways. Hence if we operate with QiQP on the product (f>ky"Ui'P and then put )/ = .v we obtain the sum of a number of terms each of which has the same deter- minantal factors as except that in any q of them u is replaced by k and p new ones of the type (aku) are introduced while p factors of the type a^ disappear and finally the factor ^^""P-* is introduced in each term. Consequently in the resulting expression there will be contained every term derived from <ji in the reverse process explained above with these definite values of p and q. We may conveniently call the transvectant of (f> and k,.^ whose indices are p, q, the order of the indices being essential, and we have the resvdt that every concomitant of degree r is the sum of a number of terms each occurring as a term in a transvectant of a form of degree r — 1 with k^": Naturally when there are different forms we have to introduce in turn a symbol belonging to each. 234. We next require certain relations that exist among the terms of the same transvectant, and to establish them we shall alter our notation for a moment. Suppose in fact that = aj^) aj^) . . . ajp') u^ u^^ ... u^ ,M, where M contains neither u nor x: In each term of the transvectant we have q of the u's replaced by k, p of the terms a J'') by (aku). We shall call two terms JV^ and aVg adjacent when jo + y — 1 of the factors affected in (f> to obtain them are common, and two cases will arise according as the remaining factor affected is an a^ or a m . In the first case we have, supposing that aj^) and aj*) are the additional altered factors in the two terms respectively, iV^i - A\ = N {(a('-)/{-M) aj^') - («W/-m) aj^'')] =N{{ai>-)ai')k) u^ - (aWa(»)M) k^}. 233-234] TERNARY FORMS 297 Now iV(a<'")a(*)ifc)is a term in where ^^ is deduced from by changing aj^'^)aj^) into (at'") «(*)«) and further ir(a('-)a(»)tt)ifc^ is a term in {({y.^, kx"}"'^'^ where <f>2 is the same as (f)^. In the second case let ic^ and it^ be the additional factoi-s altered in the • J two terms, then = —^^'{aiOjhl} and this latter is a term in where ^3 is deduced from by changing u u^ into {oiajx). Now if we call the sum of the order and class of a function its grade it is evident that (f>i, (f>2, (f)^ are each of grade less by unity than that of 0. Further between any two terms of the transvectant we can insert a number of others such that any two of the whole sequence are adjacent in our sense of the word and accordingly we have the important theorem : " The difference between any two terms of the same transvectant can he expressed in terms of transvectants of functions of loicer grade than (f) with i'^\" Thus if we consider our function <p of degree r — 1 in ascending order of grade we need only retain one term out of each transvectant that we consider — or if we please the sum of any number of terms will equally serve our piu^)ose and in particular the transvectant itself might be used. It follows at once that if a transvectant contains a single reducible term it may be neglected entirely. Again, if there be a linear relation among a number of the forms of degree r — l there will be a linear relation among the transvectants of given index formed from them, so that we need only consider linearly independent forms of degree r — l. In particular, zero forms of degree r—l can be entirely put out of account. A knowledge of the irreducible system up to and including degree r — l therefore gives us immediately all the forms ({> of which transvectants need be considered, for we have only to include the irreducible forms of degree r — l and such simple products of the others as are of total degree r- 1. We have nosv eflfecte^l our purpose of making the method at present under discussion of real value, and we proceed to illustrate it by reference to the complete system for a single quadratic. 298 THE ALGEBRA OF INVARIANTS [CH. XIII 235. Quadratics. We have here five different seta of indices for trans- vectants, namely ^ = 1,^ = 0; ^ = 0,2 = 1; ^=2,y = 0; p = l,q = \; p = 0,q^2. Consider now how far products need be taken into account, if p + q = l then all products may be neglected because only one factor is modified and hence some terms of the transvectant of a product are certainly re- ducible. If jD = 2, g' = 0, then a transvectant of ^j^o with k^^ will contain reducible terms unless the orders of ^^ and f^, ^^^ both unity, also for ^=0, ^=2, we need only consider in like manner products of two forms whose class is unity. If p = l, j = 1, we need only consider the jiroduct of two forms when one is of zero order and the other of zero class. Throughout products of more than two forms need not be taken into account. Further, in every case pure invariants of degree ;■— 1 can give rise to no new forms. 236. Single Quadratic Form. Of the first degree we have Ctx' = Ox = Cx = . . . Proceeding to the second degree we have {abu) ttxhx for p = \, q = and {abuY for p = 2, q=0 of which the first is zero. Third degree. From {abuY we get (abc) (abu) Cx (01) (abcf (02) and from Ox^ftx^ we can only get reducible forms. Now {abc) {{abu) Cx) = \ iabc) {(abu) Cx + (bcu) a^ + (abc) iCx} = l{abcfux' so that of the third degree we have only the invariant {abcf. Fourth degree. From {ahafc^ we need only consider {abd) (abu) (cdu) Cx arising from j9 = 1 and 5' = 1. This is J (abu) (cdu) {{abd) Cx + {bdc) ax + {dca) bx + {cab) dx\ = 0. For further forms we need only consider transvectants of products of powers of Cx^ and {ahif with p+q:!^2. These all contain reducible terms and hence there are no new forms so that the complete system consists of Ux^, {abiif and {abcf as ah-eady indicated. For the case of three quadratics and incidentally two see Baker, Camb. Phil. Trans. Vol. xv. CHAPTER XIY. APOLARITY {continued). 237. Apolar Conies. Two conies 8 and S' whose equations in point and line coordinates are respectively S = ax = axx + 63^2- + cxi + ^fx^x-i 4- 2gx.iXi + Ikx^x.^ = 0, and 2' = uj = ^ V + ^V + ^''"s' + 2i^'?t2«3 + 2G'usU, + 2ir'wiW2 = are said to be apolar when the invariant tv", or what is the same thing, aA' + hB' + cC + %fF' + 2gG' + 2hH' vanishes. This relation between the two conies is not a symmetrical one, inasmuch as it arises from the point equation of one and the line equation of the other; it is convenient to have an alternative name shewing the exact relation between the curves. For reasons to be explained later we shall say that S is harmonically inscribed in S', and that S' is harmonically circumscribed to S. The curves are also apolar when a'^- = 0, but in this case S' is harmonically inscribed in S. As is well known from the geometry of conies, a^'- = is the condition that there should exist an infinite number of triangles self-conjugate to S and circumscribed to S', or an infinite number of triangles inscribed in S and self-conjugate to S' — in fact the equations a^' — and ?/„'- = can be so transformed that the first has no product terms and the second has no square terms, or that the first has no square terms and the second has no product terms. The relation 0^ - = is linear in the coefficients of the equations and Ua'- = 0, 300 THE ALGEBRA OF INVARIANTS [CH. XIV hence a conic apolar to the conies Si, S.2, ... Sris apolar to any conic XiOi + X2'^2 + • • • + \r^r ^^ 0. Further if these r conies are linearly independent there are (6 — r) linearly independent conies apolar to them. In particular there is a unique conic apolar to (harmonically circumscribed to) five given linearly independent conies. The same remarks apply to p given conies 2/ = o, 2; = o,...v = o. and in particular there is a unique conic apolar to (harmonically inscribed in) five given conies. 238. Particular Cases. (i) If a^^ = represents two straight lines and Ua.^ = 0, the two lines are conjugate with respect to Ua^ = 0. If a^ = represents two straight lines coinciding in I then the line I touches the conic u^'- = 0. (ii) If Ua'^ = represents two points then these points are conjugate with respect to a^^ = 0. If Wa" = represents two points coinciding in I then the point I lies on the conic a^ = 0. All these statements can be verified immediately by using the apolar condition expressed in terms of actual coefficients or symbolically, e.g. if aj^ = v^Wx the apolar condition is which is the condition that the lines Vx = 0, Wx = should be conjugate with respect to iia'"^ = 0. 239. Ex. (i). If two pairs of opposite vertices of a complete quadrilateral are conjugate with respect to a given conic so also is the third pair. Let the conic be a^^ — o, and suppose the two pairs of opposite vertices are given tangentially by The general equation of a conic inscribed in the quadrilateral is then \UpUp'+HUgUg'=0, and since UpUp' = and UgU^=0 are both apolar to ax^=0 it follows that every conic inscribed in the quadrilateral is apolar to ax^ = 0. But the third pair of opposite vertices is one such conic, hence these remaining vertices are conjugate with respect to the given conic. We shall call such a quadrilateral a quadrilateral harmonically inscribed in the conic 0x^ = 0. 237-240] APOLARITY 301 Ex. (ii). Four conies have in general one common harmonic quadri- lateral. Let the conies be S^, S^, S^, S^, then the apolar system is of the type consequently the apolar conies in general all touch four fixed straight lines. The opposite vertices of the quadrilateral formed by these lines taken in pairs constitute conies of the apolar system, and hence pairs of opposite vertices are conjugate with respect to each of our four conies. Hence the quadrilateral is harmonically inscribed in each of the given conies. Ex! (iii). A triangle ABC and its polar triangle with respect to a conic are in perspective. For if the polars of B and C meet the sides CA and AB respectively in Q, R, and the line QR meet BC in P, then the quadrilateral formed by BC, CA, AB and the line PQR has two pairs of opposite vertices, viz. {B, Q), (C, R) conjugate with respect to the conic ; therefore {A, P) are conjugate with respect to the conic, or the polar of ^ meets BC m. P. Thus the polars oi A, B, C meet the opposite sides in three collinear points, and they therefore form a triangle in perspective with ABC. Ex. (iv). If u^W = 0, «/^) = 0, Mx<^) = 0, Wx(*) = be the sides of a quadrilateral harmonic with respect to the conic S=0, then we have S=Xiui\^+\u(%^ + \su(%^ + XiU(%^. For let two pairs of opposite vertices be (pp') and (qq') ; then apolar to the conies (pp) and (qq') we have the five conies v.(i):,2 = 0, ^(2)^2=0, id%^ = 0, u(%^ = and S=0. But the first four are linearly independent and hence S is a linear combination of them. 240. Some interesting applications can also be made to the metrical geometry of conies. In fact, suppose that the tangential equation of the circular points at infinity (I, J) is <f) = tiy- = 0. Then a conic apolar to (f> has /, J for conjugate points and is therefore a rectangular hyperbola. Again, the tangential equation of a circle whose centre is j9 is of the form Up^ = \<^ where \ varies with the radius. 302 THE ALGEBRA OF INVARIANTS [CH. XIV If a circle G be apolar to a conic S = w,^ = then the director circle of the conic cuts the circle C orthogonally. Use rectangular Cartesian coordinates, and let the equations of the conic and circle respectively be AP + 2Elm + Bill" + 2GI + 2Fm +0 = 0, and a^^ + 2/2 + 2gx + 2/3/ + c = 0, so that we have A+B + 2gG + 2fF+cG==Q. But the equation of the director circle of the conic is G{x' + y'')-'lGx-^Fy + A-\-B = 0, and this cuts the given circle at right angles if ^ G ^.F A + B _ le.ii 2gG + 2fF+cG + A+B = which is precisely the condition of apolarity. The director circle of a conic inscribed in a triangle cuts the self-conjugate circle oHhogonally. For since the self-conjugate circle has the triangle for a self-conjugate triangle and the conies are inscribed in the triangle, each of the conies is apolar to the circle. Hence their director circles cut the self-polar circle at right angles. Or thus, — the system apolar to the inscribed conies is of the form M^x + mx^ + vi'^^ = 0, where p^ = 0, qx = 0, rx = represent the sides of the triangle. By suitably choosing \, fj,, v this equation may be made to represent a circle, and from the form of its equation it is the self-polar cu-cle of the triangle. The locus of the centre of a circle which has two fixed pairs of conjugate lines is a rectangular hyperbola. In fact, suppose the lines are px = 0, qx = O1 rx = 0, Sx = OJ 240-241] APOLARITY 303 The system apolar to the tangential system of conies having these two pairs of conjugate lines is ^PxS'x + f^r^S:, = 0. There is one value of the ratio X : /* for which this represents a rectangular hyperbola. Let >Sf = be the rectangular hyperbola in question and let U,f = \^ be one of the circles. Then B is apolar to ^, because <S = is a rectangular hyperbola, and as it is apolar to Up^ = \(f) it is apolar to i/,/. Hence the point jj must lie on S and therefore S is the centre locus of the circles. In general when a rectangular hyperbola S is apolar to a circle 2 the centre of the circle lies on the rectangular hyperbola. 241. Apolar Curves in general. The two curves whose equations are and </> = Ua^ = OJ are said to be apolar when the form a^ar^^~'^, which we denote by •v|r, is identically zero. Except for a numerical multiple we have ' \dUxOX^ 011^0X2 ou.oxj The following are analogous to theorems on binary forms. I. The form <j) is apolar to any polar of f whose order is not less than n. For {a^'''~''ay'-, iC}"''' = a^a^-^-'^ay"" and this is zero as we see by polarizing the identity «»''«*"'"'' = r times with respect to y. 304 THE ALGEBRA OF INVARIANTS [CH. XIV II. The fovTn f is apolar to any form which contains (f) as a factor and whose class does not exceed m. For if <^' be any form of class n we have {/, ^,^'|o.»+.' = |(y; ^)o.n,^'|o..' = 0, since (/, </>)"''' vanishes identically. Hence / is apolar to 0<^'. We have supposed that m'^n hitherto. Exactly similar remarks apply to the case in which n > m. The search for the forms of given class (n) apolar to a given form / is facilitated by the fact that the necessary and sufficient conditions for <f) are that it should be apolar to every (m — w)th polar of/. For an (m — n)th polar is and this is apolar to Wa" if aa^ay"'-'' = 0. But if this relation be true for all values of y then the form ^ is apolar to/. 242. Ex. (i). A ternary cubic has three linearly independent apolar conies. For the first polars of the cubic are linear combinations of dx-^ ' dx^ ' dx^ which are three linearly independent quadratic forms. Hence there are three linearly independent conies apolar to all first polars and therefore apolar to the cubic itself. Ex. (ii). A ternary quartie has an apolar conic mdy when the deter- minant of the coefficients of its second differential coefficients vanishes. For an apolar conic must be apolar to all second polars and they are linear combinations of 82/ a^ a^ _^f ay ^f dx^^ dx^^' 8-^3^' 8.^23.^3' dx^dxi' dxidx.2' In general there is no conic apolar to each of these six, but there will be an apolar conic if the six be not liuearly independent, i.e. if the determinant of six rows and six columns be zero. 243. G-eneral Theory of Curves which possess an Apolar Conic. By using suitable coordinates the analysis of ternary forms apolar to a given conic may be reduced to that of binary forms. 241-243] APOLARITY 305 Suppose that the fixed conic is x-^x^ — xi = or in line coordinates 4iUiU^ — u^ = 0. There is no loss of generality in taking the equations in this form, because by suitably choosing the triangle of reference, the equation of a proper conic can be always reduced to the form 13 ^~ '* — Thus we may take for the parametric representation of points on the conic X^ = Vi, X2 = I'l ^'2 > ^3^^ ^2"> and we shall call this point (^1, x^, x^) the poiut v. If the line u^^^O meet the conic in the points (\ yx) the quantities X, /m are given by UiVi' + U2V1V2 + u^v^ = so that Ui = X.2/A2 «2 = - (^1/^2 + ^2/^1) ■ (A> Us = Xi/ii except for a constant factor. Hence we may regard the quantities as the coordinates of the line, and a homogeneous relation of order m connecting the u's becomes a homogeneous symmetrical relation of order 2m between the X's and /x's. Thus any symmetrical relation between X and fi is equivalent to the tangential equation of a certain curve whose class is one-half of the order of the given relation. The coordinates of the tangent at the point v are and hence the points of contact of the tangents from the point x to the curve are given by x^vi — Ix^V-^Vo + x^v^ — 0, so that we may take X\ = Xi/^i j Xs = Xa/ia j G. & Y. 20 306 THE ALGEBRA OF INVARIANTS [CH. XIV Consequently a homogeneous symmetrical relatio. the \'s and /i's of order 2n represents a curve of order n. '^' ^ . given a symmetrical relation we therefore deduce two cu ° equations from it, one in line coordinates and the other in poii ' coordinates. The curves represented are reciprocal with respecb to the fundamental conic because, taking X,, /j, to be fixed, the line u given by (A) is the line joining them, and the point a; given by (B) is the intersection of the tangents at A, and /i, i.e. the pole of the line u. This method of representing a point by the parameters of the tangents drawn from it to a fixed conic and a line by the parameters of the points in which it meets the conic was practically used by Hesse and first explicitly used by Darboux. 244. By means of this system of coordinates we can readily find all curves apolar to the given conic. I. Suppose that w^" = is a class curve apolar to the conic, then we have (72' - 7173) V = 0. Thus (72^ — 7173) multiplied by any function of the 7's is zero if it be interpretable, hence this symbolical expression must be zero, and we may write 7l = «2', 72=-aia2, 73 = «lS and our equation is The a's are now the only symbols used, and it is clear that as any expression of degree n in the 7's represents an actual quantity, any expression of degree 2n in the as is an actual coefficient, or in other words the as are the symbols of a binary form of order 2n. On introducing the Vs and fi's our equation becomes [a^%ifi2 + Oitta (\i/*2 + ^2/*i) + ai^A-i/ii}** = 0, or {(OiXi + ttaXj) (tti/Ai + a^fi^Y = ^' that is finally ax^a/ = ^- 243-244] APOLARITY 307 The binary 2n-ic of wliich the as are symbols has an important significance, for if we make X. = //., the line Ux is the tangent to the fundamental conic at the point X; consequently the equation a^' = gives the parameters of the points of contact of the 2n common tangents of the apolar curve and the conic. Conversely, when the equation ax^" = is given, the equation aA."a/' = is uniquely determined by polarizing, and hence we have the theorem that a class curve apolar to a conic is uniquely deter- mined when its common tangents with the conic are given. By proving this theorem from first principles, and then observing that a^^'a^'' = 0, or its equivalent {a^'Ui — a-ia^iio, + o.^u^''^ = 0, certainly represents a curve apolar to a conic, we can shew that any apolar curve may be*reduced to the form ttA^a^^' = 0, without using a parametric representation of the symbolical equation II. Suppose that the curve c/' = of order n is apolar to the given conic 4wiW3 — u^ = 0, then we must have (4ciCs-c,»)c^"-^=0, and reasoning as before, we have * See Schlesinger, Math. Ann. Band xxii. 20—2 308 THE ALGEBRA OF INVARIANTS [CH. XIV We may now use the parametric representation Ci = ttiS C3 = fl2 , and the equation of the curve becomes (ai^nci + la-fl.^^ + a^^x^Y = 0, or introducing the X's and /a's {a^Xifii + a-^a^ (Xi/Ua + Xojx^) + a-fXif^o]'^ = 0, i.e. a^'^a^'' = 0, and as before the a's are the symbols of the binary 2n-ic a^''", which, equated to zero, gives the points of intersection of the apolar curve and the conic. Example. To find the conic apolar to x^x^ — x.^ = Q ivMch touches the tangents to this conic at the points given hy v^^-v.^ = 0. Here a\2n = is Xi*-X2* = 0, thence ax"a/u,"=0 ("B-xT'V-V). or XiVi'^-X2V2^=0. On using the substitutions the equation becomes u^ — u^ = 0, or («3+Mi)(W3-«l)=0. so that the conic consists of the two points (1, 0, 1) (1, 0, — 1), and in fact it is easy to see that these points are conjugate with respect to the conic. 245. Theorems on conies apolar to the fundamental conic. The equation of a conic apolar to x^x^ — x-^^Q, and touching the tangents at the points given by a/ = 0, is equivalent to a^^a^^O, and hence to m/ = 0, where Uy = a^a^. Now suppose that A, B, C, D are four points (X, fj,, v, p) on 244-245] APOLARITY 309 the fundamental conic, and that the lines AB, CD whose equa- tions are Vx = and w^ = 0, are conjugate with respect to the apolar conic Uy^ = 0.' Then since the condition of conj ugacy is VyWy = and Vy = iif^a^ , Wy = a^a^, we have a^^a^ap — 0, that is the quartic giving \, fi, v, p is apolar to a/ = 0, This is one of the simplest geometrical representations of forms apolar to a given form. From the symmetry of the result, we see that each pair of opposite sides of the quadrangle ABGD are conjugate with respect to the apolar conic. Thus there is an infinite number of quadrangles inscribed in the fundamental conic and harmonic to the apolar conic ; the four vertices are given by fo^ms apolar to the form a,' = 0. Now if X, yu-, V be chosen so that p is arbitrary, any line through A is conjugate to EC, so that A is the pole of BG, and hence ABC is a self-conjugate triangle of the apolar conic m/ = 0. In this case the cubic giving \, /x, v is apolar to a^^ = 0, and therefore to every first polar of this form ; hence there is an infinite number of triangles inscribed in the fundamental conic and self-conjugate with respect to the apolar conic, and their vertices are given by the singly infinite number of cubic forms apolar to a^* = 0. Next suppose that the linear factors of the quartic giving A,, /A, V, p are Ij, m^, iij, i\, then where X, M, K, R are independent of r. By polarizing we obtain the identity a^a,^ =-- L {kl,y -H M {vnm.y + N{n^n,y + R {r^r^)\ where f, r^ are any two points on the conic. Now by means of the usual substitutions "l = 1^2-^2, U^ = - (^1^2 + ^2'7l), "3 = 11%, the left-hand side becomes Wyl 310 THE ALGEBRA OF INVARIANTS [CH. XIV Consider next the term l^l^. Since ^1 = X2 ^■iid Z2 = — Xi, this becomes or Xi^Ui + \1X2W2 + Xa^Ws , and Xj^, X1X2, Xg^ are the coordinates of ^, so that hh — ''■''A > where w^ = is the tangential equation of A. Hence we have m/ = Xii/ + Mub^ + Nuc" + Rub'', and the conic is represented as the sum of four squares. In particular, ii A, B, G he the vertices of a self-conjugate triangle, we obtain in like manner V = Lua^ + Mub^ + Nu(?. These results are well known and easily obtained otherwise, * but the methods here used may be applied with equal success to more difficult problems as we shall presently shew. Exactly the same reasoning applies to a conic c^ apolar to 4iVriUz — U2 = 0, and now the triangles are circumscribed to the fundamental conic and self-conjugate to the apolar conic. 246. Condition of apolarity of two conies apolar to the standard conic. Suppose the two conies are Cy? = 0, Wy- = 0, that the first meets the standard conic in the points a^* = 0, and the second touches the tangents to the standard conic in the points 6a^ = 0. Then we have Cx = axa^ and Uy = bj)^, and in particular C2 = 2a^ar, [ , 72 = - 6162 [ , C3 = ai ) 73 = 61^ i hence c^ = a^h^ + a^h^^ - 2a^a.hJ)2 = {ahf, 245-247] APOLARITY 311 and the conies are apolar when Cy = 0, that is when (aby = 0, or when the two binary quartics a/,* and 6x* are apolar. This gives another simple geometrical representation of apolar quartics. 247. To many of the theorems developed for conies there are analogues for all curves possessing an apolar conic. For brevity we introduce a definition. If the equation of a curve of order m be J {Xi, X.2, 0C2,) = U and/ can be written as a linear combination of the forms u^wm^ r = 1, 2, ... w, then the n lines Ma;*'"' = are said to form a conjugate n-line with respect to the curve. In like manner if the tangential equation of a curve of class m be and ^ can be written as a linear combination of the forms w^J^ r=l,^,...n, then the n points Ux =0 are said to form a conjugate n-point with respect to the curve. Suppose then that the curve is apolar to the fundamental conic 1 *^3 """ 2 — ^> the equation may be written where ax^" = gives the 2n points in which the curve meets the conic. If aK^^ be apolar to the form whose linear factors are Pk''\ Pk'\ ... Pk^'K 312 THE ALGEBRA OF INVARIANTS [CH. XIV then a\^ can be expressed as a linear combination of so that af'a^^ is a linear combination of and hence just as in the case of the conic it follows that the tangents to the conic at the points given by i.e. by 6/ = 0, form a conjugate r-line with respect to the curve whose equation can accordingly be expressed as a sum of r ??th powers. The above will suffice to indicate the general principles which we shall now apply to the ternary cubic and quartic. 248. Ternary Cubic. A cubic curve, as we have seen, § 242, always possesses an infinite number of apolar conies. Take the fundamental conic for one of these and let c^' = be the equation of the cubic. This may be written and meets the conic in the points given by ax« = 0. This binary sextic has three linearly independent second polars, and therefore a singly infinite number of apolar binary quartics but not in general an apolar cubic. Hence a ternary cubic may be written in an infinite number of ways as the sum of four cubes for each apolar conic it possesses, but not in general as the sum of three cubes for an arbitrary apolar conic. The condition that the binary sextic may have an apolar cubic is that the determinant formed by the coefficients of its third differential coefficients may be zero, i.e. that any four third polars may be linearly dependent. 247-249] APOLARITY 313 This condition is (aay {a'a'J {a"af {aa'y {a a" J {a" a'")- = 0. (§ 187) Now ai^ = Ci, 2a^a2 = c.2, hence {aa) {a' a") {a" a) = Hcc'0 i a'? a/'a," a\ , and the condition reduces to (cc'c") (c'c"c'") {c"c"c) {c"'cc') = 0. This is an invariant of the cubic and hence however we choose the apolar conic we cannot reduce the cubic to the sum of three cubes unless a certain invariant of degree four vanishes. 249. Ternary Quartic. Here there is no apolar conic unless the six second differential coefficients are linearly de- pendent, i.e. unless a certain invariant called the Catalecticant vanishes. Now if the quartic can be written as the sum of five fourth powers it must have an apolar conic, because a conic can be chosen apolar to any five fourth powers — in fact we have only to describe a conic touching the five straight lines represented by the linear forms. Hence in general a ternary quartic cannot be expressed as the sum of five fourth powers. But if the catalecticant be zero there is an apolar conic and, taking it for a fundamental conic, the equation of the quartic may be written where ai^ = gives the points of intersection with the conic. Now a singly infinite number of quintics can be found apolar to a binary octavic, hence in this case the quartic curve has a 314 THE ALGEBRA OF INVARIANTS [CH. XIV singly infinite number of conjugate five-lines, and all such lines touch the apolar conic. 250. We shall conclude this chapter with a brief account of the class of invariant forms known as combinants, confining ourselves to binary forms. An invariant or covariant of any number of binary forms /l. Jly ••• Jr of the same order is said to be a combinant if it be unaltered, except as regards a factor independent of the forms, when each form / is replaced by a linear combination of the type lifi + kf2 + •'■ +lrfr, in which the I's are constants. For example in the case of two binary forms we have (hfi + kA, m^fi + m^A) = {lm){fj,\ so that the Jacobian of two binary forms is a combinant. For the sake of brevity we shall deal with the combinants of three binary forms /a = &o^i" + n&i«i"~'«o + . . . + hnX^"" J 3 ^ Cq Xi "T nCi Xi X2 + . . . "T Cji x^ . A combinant is not only unaltered by a linear substitution efiFected on the variables, but also by a linear substitution of the type a/ = liar + rriibr + n^Cr br = hdr + m.ibr + Tl^Cr Cr = ^ttr + Wfs^r + ^hCr effected on the coefficients. Regarded as a function of the coefficients the combinant is therefore an invariant of the linear forms dr^i + K^i + Cr^S, r=l, 2 n, because if we put ^i = li^i +^2 +h^3' etc. 249-250] APOLARITY '^^^F 315 we find a/ = lia,. + niibr + n^Cr etc. which are the substitutions above. Hence, by the fundamental theorem on the symbolical representation of invariants, a combinant, as far as the coefficients a, b, c are concerned, is a rational integral function of determinants of the type 'p hq br I Cp Cq Cr A like result applies to any number of binary forms. Thus for example for two quadratics &o^i^ + 2biXiX2 + b2xi, a combinant is a function of {a^by - aJ}o), (a^b.^ - a^b^), {aoh - a^h), as far as the coefficients are concerned. But the Jacobian is (ao^i — rti^o) ^1 + («0&2 — ^2^0) Xl ^2 + («1^2 — CbJ^l) ^2^ and hence a combinant is a rational integral function of the coefficients of the Jacobian and the variables. Hence any combinant is an invariant form of the Jacobian, and therefore the complete system of combinants in this case consists of the Jacobian and its discriminant — the latter is equivalent to the resultant of the two original forms. It is easy to form any number of combinants of two binary forms, for (i) An invariant or covariant of a combinant is itself a combinant, since it is manifestly an invariant form £lnd further involves the coefficients of the original forms in the manner peculiar to combinants. (ii) Let fi and/*2 be two binary forms, /„ an invariant form of /i and the corresponding form for Xj/j + X0/2. Xi"»/o + ^r-'^Ji + . . . + Xa'^/m , then an invariant of this expression considered as a binary form in (Xi, X2) is a combinant ofy*! and /g. 316 THE ALGEBRA OF INVARIANTS [CH, XIV For such an invariant is unaltered when we effect a linear substitution on the x's because each / is an invariant form ; and it is unaltered when we effect a linear substitution on the coefficients because it is an invariant of the form in X. written above. 251. Combinants naturally occur in the discussion of rational curves as we shall now shew. Suppose such a curve is parametrically represented by ^, = K- =/, ^^ = c,^'=fs, (cf§196) then the curve is unaltered by a linear substitution effected on the x's since its equation is found by eliminating the x's. Now if a set of points on the curve be defined by some projective property the equation giving their parameters is derived from /i, /g, fa in a definite way, hence if by means of a linear substitution/i./a^/s become// .//./a' the transformed equation for the parameters is derived from//,/',/' in the same way as its original form was derived from/, /a,/. Thus if the equation be C = it follows that C is a covariant of/,/, /s. Next, keeping the parameters fixed, to change the triangle of reference we replace ^i, ^2, fs hy linear functions of themselves, so that/,/,/ are replaced by linear combinations of the form hfi + 4/ + hA- Now the equation giving the parameters of the set of points must be independent of the triangle of reference, for such points depend on the curve itself, and the parameters of every point of the curve are unchanged when we alter the triangle of reference ; hence G is not only a covariant but a combinant of the forms /, /, /, and the rational curve is the natural geometrical representation of the system of combinants. The curve can be equally well defined by the system of forms apolar to /, /, /, because these determine /, /, /, and the projective properties of the curve are also given by the combinants of the apolar system of forms. 250-252] APOLARITY 317 We are therefore led to the theorem that the combiuants of two apolar systems of forms are identical, and in fact a rigorous algebraic proof of its truth will be found in Meyer's Apolaritdt, § 11. As an example the reader may verify that in the quartic curve gi =' Clx > 6 2 ^^ '^x > S3 ^^ ^a; > the points of inflexion are given by (he) (ca) (ah) a^hiCx = 0, and that, if d^;* and e^* be two forms belonging to the apolar system, they are also given by {de)dx^ex' = 0. The first equation follows from the ordinary methods of the differential calculus — the second from the fact that the conditions of collinearity of four points are easily expressed by means of the apolar system ; if A, be a point of inflexion, and fj, the point in which the inflexional tangent meets the curve again, we have d^^d|J, = 0, e^^e^ = so that /jl may be eliminated. The full discussion of the theory of combinants would lead us too far from the methods of the present treatise, and accordingly Ave shall content ourselves with the explanation of a "translation- principle" connecting the combinants of binary forms with the covariants of ternary forms. 252. It will be convenient to change the notation and to suppose a rational curve given by ^j = ttjiCji + nb^.%\''-^cc,^ + + hx^''] ^2 = ^2^1" + nh-iCc^''-^ 0:2+ + k^x^^'i- ^3 = as^i"" + tAx^^'-^x^ + + Ic^x^""] Consider the problem of finding the locus of the point of intersection of two straight lines which meet the curve in two sets of yi points given by binary forms for which a certain combinant is zero. Let the two lines be Wf = 0, v^= 0, and denote by ^ their point of intersection. The two binary forms are aw^i" + nbuXi''--^x.^ + . . . + A;„a;/' and a^x^"^ + nh^x-^"^-^ x^-^- ... + ky x^ , 318 THE ALGEBRA OF INVARIANTS [CH. XIV and any combinant is a function of determinants of the type Clu Om Now UuK -a^bu = (ab^), hence the equation of the locus is found by changing (ab) in the expression of the vanishing combinant into (a6f ). For example if two lines meet the cubic curve 1^1 = aiX^^ + 2)b^x^X2 + ^c^x-^x^ + d^x^ etc. in two apolar sets of points we have {audy, — a^dy) — 3 {buCr, — b^Cu) = or. {ad^)-S(bc^) = 0, and hence the locus of their common point ^ is a straight line. It is evident that if J. be a point of inflexion then the tangent at A and any line through A satisfy the conditions of the problem, so that all the points of inflexion of the curve lie on this straight line. As a second example let us find the equation of the cubic. Here the two straight lines meet on the curve and the vanishing combinant is the \ eliminant of the two binary forms. Following Bezout's method, the eliminant of pxi^ + qx^x^ + rx-^x^ + sx^ and "p'x^ + q'x^x^ + r'x-^x^ + s'xi IS = 0, W) (K) (p«') (_pr') (ps') + {qr') (qs') (ps) {qs') (rs') and hence making p = au, q = 36m etc. the equation required is 3(a6|) 3(ac^) (ad^) =0. 3 (ac^) (ad^) + 9 (bc^) 3 (bd^) (ad^) S{bd^) S(cd^) It is clear that a similar method applies to the curve of the nth degree. CHAPTER XV. TYPES. 253. It was proved in § 35 that the effect of operating with on a covariant <I> of a simultaneous system of binary forms, which includes ag." and bx^ where a^"" = (Ao, ^1, ... AnJ_oc^, os^T bx'' = (Bo,B„ ... Bn^x„x.^», is itself a covariant of the system. All covariants thus obtained from <t> are said to be of the same type as <I>. In other words two covariants are said to be of the same type if one of them is obtainable from the other by means of operators of this kind. For example the invariants of two quadratics /, cf) are all of the same type. It should be noticed that this connection between two co- variants is not necessarily reciprocal; two covariants <I>i, <I>2> where <E>2 is obtainable from <I>i by operators of the required kind, are of the same type, even if ^j is not so obtainable from ^2- Thus if F(a, a', ...) is a simultaneous covariant of a system of quantics which includes /=««'* = aVS (f> = ba;^, and if F is of the second degree in the coefficients of / but does not contain those of (j>, the covariant F(a, b, ...) + F{b, a, ...) = (<|,|.) F {a, a, ...) 320 THE ALGEBRA OF INVARIANTS [CH. XV is of the same type as F{a, a', ,..) ; but here F{a,a,...)={^^F{a,h,...) and we see that F{a, a', ...) and F{a, b, ...) are of the same type. It will be seen in this way that two covariants ^i, <E>2 may each be of the same type as a third covariant <I>, although neither <J>i nor <l>2 is obtainable from the other by an operator of the kind considered. In view of this the further statement is necessary that covariants which are each of the same type as a third covariant are (by definition) of the same type as each other. 254. Every covariant of degree m, of one or more quantics, is of the same type as a covariant which is linear in the coefficients of each of m quantics — the number of quantics in the system being, of course, increased if necessary. Any such representative covariant is, for convenience, called a type; a type is then a covariant which is linear in the coefficients of each of the quantics concerned, it being understood that these are not special quantics of the system and that the word type is used in a purely formal sense. Thus for three quadratics (§ 139 a) {ab) (be) (ca) is an irreducible type, and furnishes only one invariant of the system ; (ab)^ is also an irreducible type and furnishes six invariants. It should be noticed that if fi,f2,fz are the quadratics, and the invariant i'^i.iifzf' is not of the same type as (a&)^, because J^^^ ^^ "^^ ^"^ '^^ ^he fundamental quantics of the system. Consider the covariants of a simultaneous system of binary forms of the same order. When the number of binary forms is indefinitely increased, the number of irreducible covariants will also be increased without limit ; in fact the number of irreducible covariants belonging to any one type will be indefinitely increased. The question arises — does the number of irreducible types increase indefinitely too ? This question has been answered in the negative 253-256] TYPES 321 by Peano*. Peano's theorem is the following: Every type of a system of binary n-ics which does not furnish irreducible covariants for a system of n n-ics is reducible, with the single possible ex- ception of the invariant type A^ Ai ... An Bo -Bi ... Bn Ko Ki ... K,, where {A^, A^, ... A n\x^ , x^Y (Bo,B^, ...Bn\x„x,y {Ko,K^, ...Kn\x„x^f are ?i + 1 n-ics. But if this invariant is reducible, all types are reducible which do not furnish irreducible covariants for n — \ n-ics. A proof of this theorem is given in the next chapter. 255. As was pointed out in § 21 there are two principles by means of which the reduction of a covariant has to be attempted, viz. : (i) The fundamental identities (6c) (ad) + (ca) (bd) + (ab) (cd) = (be) a^ + (ca) b^ + (ab) c^ = 0. (ii) The fact that the interchange of two symbols which refer to the same quantic does not alter the actual value of a symbolical product. To effect the reduction of a type the first of these two principles must alone be employed. 256. The quadratic types. The quadratics will be denoted, as usual, by ^x'j ^x'} ^x'y • " • * Atti di Torino, t. xvii. p. 580 (1881). See also Jordan (Liouville, 1876, 2 S6r. III.), who proved that the number of irreducible types belonging to any simultaneous system of forms, the order of each of which is less than some fixed number n, is finite. G. & Y. 21 322 THE ALGEBRA OF INVARIANTS [CH. XV For invariants the only symbolical products to be considered are {ahf ^ {ah) (6c) {ca) {ah) {he) {cd) {da) The first two of these are irreducible, for the fundamental identities give us no relations by which we may reduce them. The other invariant types are all reducible. For 2 {ah) {he) {cd) {da) = {ahf {cdf + {hcf {daf - {acf {bdf ; operate on this identity with A A then {ah) (he) {cd) {de) + {eh) {he) {cd) {da) = - {cdf {ah) (he) - {hcf {ad) {de) + {hdf {ae) {ce). But {ah) (be) {cd) {de) - {eh) {he) {cd) {da) = (be) {cd) {dh) {ae). Hence 2{ab){hc){ed){de) = {he) {cd) {dh) {ae) - {cdf {ah) {he) - {hcf {ad) {de) + {hdf {ae) {ce). By means of these two identities all the invariant types of degi'ee greater than three are at once reduced. Now any covariant of a system of quantics of even order must be itself of even order (§ 20) ; hence any covariant type of the quadratic may always be obtained by replacing one or more letters in the symbolical expression for some invariant type by the variable. For example, from {ah)- we obtain ax' on replacing hi by — iCg and bs by a^i. Hence the irreducible covariant types are a/, {ab)axhx. The quadratic has then only four irreducible types (compare §139 A), {ahy, {ah){be){ca), ai, {ab)axbx. 256-257] TYPES 323 The second is in fact the determinant type referred to above, for as has already been pointed out, (ab) (be) (ca) = — ' ai^ aittg (^2^ b,^ bA V 257. The cubic t3rpes. It is possible to obtain the complete system of types for binary forms of a given order by a method almost identical with that of Chapter vi. for covariants of a single binary form. The reductions in this method are generally very difficult to obtain. The cubic types, however, can be thus obtained simply. It is thought unnecessary to go through the general argument, the alterations in Chapter VI. to meet the case of types being mainly verbal. It should be noticed that the finiteness of the complete irreducible system of types could thus be demonstrated. Let a^^, bx\ Cx\ ... be the cubics. The symbol F will be used to denote any one of them indifferently. The types of .degree two are {ab)ax^b^ = J, {ab)- a^bx = H, (aby. Consider first the types of grade unity. These all contain a factor (ab), and hence are terms of transvectants of J with types of grade not greater than unity. In fact any such type is a term of a transvectant of the form {J,J,,..Jr, F,F,...F,)\ where the bar over the left-hand member indicates any type obtained by convolution from the product there written down. It follows at once that every type of unit grade can be expressed as a sum of numerical multiples of such transvectants. Now by §§ 74, 75 any type obtained by convolution from Ji, J^, ... Jr is of grade two at least. Hence the only irreducible types of unit grade are expressible as transvectants of the form (JJ,...Jr, F,F,...F,)'. If \ = 1 this is clearly reducible — for J" is a Jacobian. If \ > 1, this contains a terra of grade two. Therefore the only irreducible type of unit grade is J. 21 2 324 THE ALGEBRA OF INVARIANTS [CH. XV Now the irreducible types of the quadratic H are Hence the tj^es of grade two are expressible as transvectants of the form {H,H,... H,K,K,... Kp, F,F,...FyJ,J,... J.f or of these multiplied by invariant types. In the first place we notice that K and J are Jacobians and hence we may suppose that neither yS nor B exceeds unity. We have the following types to consider : {H,F), {H,F)\ (H,H,,Fy, {H,HA,F,F,y {H,J)\ {H,H„Jy, {HJI,,jy (K,Fy, (HK,Fy, (H,H,K,F,F,y, (K,jy, {HK,Jy, {HK,jy. Of these {H^H^H^, F^F^y contains the term («! hy {aAy (ashy (aicO (61 Ci) (a^c^) (hc^) (asd) (63C2) = (aA) (61 Ci) (CiOi) . (a.262) (&2C2) (CjOa) . {a^bsy (aA) (a^h) (OsCi) (63 C2) =i (cha,y (aAy ((hc,y ; Aa,y (hAy {hxc,y \(aAy((hbi)(,aA)(chC,){bsC,),...(^71). {c,a,y {cAy {c^c,y This is a sum of terms obtained by convolution from products of four types H, and hence is reducible. In exactly the same way the type {H^H^KyF^F.^y may be reduced. The type (H, Jy contains a term iaby (be) (ad) (cd) c^^d^ = - i {oh) {(aby (cdy + (bey (ady - (acy (bdy] c^d^, which may be expressed in terms of the type (H^H^) and reducible forms. The type (K, jy contains the term (H,H,)(H,c)(H,d)(cd)C:,d^ = - i {(H,H,y (cdy + (H,cy (H,dy - (H,dy (H,cy\ c^d^, and hence is reducible. The type (HjH.^, Jy contains the term (Hu (H^, Jy) = 2(H„ (H„ H,)) + reducible terms, and hence is reducible. In the same way the types (HiH,^,jy, (HK, jy, (HK, jy may be reduced. 257] TYPES 325 The type {K, FY = (H,H,) (H,F) {H,F) F, = {H,Fy (H„F) H,, - {H.Ff {H,F) H,, ; and 2 (H,H„ Ff = {H,Ff {H,F) F^ + {H.jy (H,F) F^. Hence both {K, Ff and {H^H.^, F)- can be expressed in terms of the type {H,F)"'{H,F)F, = L^. Lastly, (HK, Ff contains the terra (aih,)- {a^Ky (asbsf {h^a^) {(hc) {a^c) {h^c) b^x = ((«! hy (O'shY (bi a^) (a, c) (agc) (bsc) a^i, b^i^. Now the left-hand member of this transvectant is a type of degree six and order two ; looking back at the possible types of this order we see that it must be of the form The second and third of these terms contain invariant factors, and can therefore only lead to reducible terms in the above trans- vectant. Also 2L,L, = 2 (aby (ac) (be) c, . {dey (df) (ef)/^ = (ab)ide)Cxfx (ady {aey {a/y (bdy (bey (b/y (cdy (cey (cjy = S (H,H.;)' H, + ^ (H,H,) {HA) H,,H,, = S (HiH^y H3. The type (HK, Fy is thus reducible. The complete system of irreducible cubic types is then (aby, (aby (be) (cdy (da), (aby (be) (cdy (de) (e/y (fa), (aby (ac) (be) c^, (aby (edy (ae) (be) (ee) 4, (aby aj)^, (aby (be) (edy a^d^, ax, (aby (be) a^ex, (ab) a^bxy there being ten types in all. The system of irreducible concomitants for two cubics may be obtained from the system of types or else directly, they will be 326 THE ALGEBRA OF INVARIANTS [CH. XV found in the works of Clebsch and Gordan, The syzygies between them have been obtained by von Gall {Math. Ann. Bd. xxxi.). 258. Perpetuants. The irreducible seminvariants (§ 32) of the binary form of infinite order are called perpetuants. The complete system of perpetuants for one binary form of infinite order has been obtained by Macraahon* and Strohf. The system is, of course, infinite in extent, but the individual members of it have all been identified. The complete system of perpetuants for any simultaneous system of binary quantics of infinite order was obtained by Macmahon;}:; and in particular the perpetuant types may be at once obtained from this paper. The method by which these results were obtained, does not fall within the scope of this book. The results have been obtained more recently by means of the symbolical notation which has been here developed; and this investigation § we shall follow. 259. A covariant is completely defined when the determinant factors in its symbolical expression are known ; it will be convenient to use this part of the symbolical expression only. In dealing with forms of infinite order, it must be remembered that the complete expression for a covariant contains each of the factors a^, h^, ... raised to an indefinitely high power. The identity (6c) ax + (ca) h^ + {ah) Ca; = may now be written {he) + {ca) + {ah) = 0. By means of this identity any factor (6c), in a covariant, which does not contain a may be replaced by {ac) — {ab), i.e. by factors which do contain a. Thus all covariant types may be expressed in terms of those which are of the form {ahf{acY{ady... where a is any one of the letters chosen at will. * Proc. Lond. Math. Soe,, vol. xxvi. See also Avi. Journal, vols. vii. viii. t Math. Ann., Bd. xxxvi. t Camh. Phil. Soc. 'Trans., vol. xix. pp. 234 — 248. § Grace, Proc. Lond. Math. Soc, vol. xxxv. 257-261] TYPES ^^^ 327 The types of this form are all linearly independent, for no linear algebraical identity can connect their symbolical expres- sions. Hence if all reducible types were expressed in terms of types of this form, we should be able to write down the perpetuant types. It must be remembered that a is a perfectly definite quantic of the system. Further that the remaining quantics concerned in any particular covariant will be considered in a particular order determined beforehand. 260. Consider the types of degree three ; if w be the weight, we know that {hcY = {(«c) - {ah)]"" = {acf" - w {ah) (acy-' + + (- 1)^ («&)«'. Hence the covariant (ab) (ac)'^~^ is expressible in terms of reducible covariants and of covariants in which the index of (ab) is greater than unity. Hence all perpetuant types of degree three are expressible in terms of the types (ab)^{acy, \<t:2, fi^l. It should be noticed that of the three quantics concerned any one may be chosen to correspond to a, b or c respectively. Further, the only reducible covariants of degree three and weight w are represented by (6c)% (aby, (ac)"", and hence the seminvariants (aby {acY (X. -"^ 2, /t -^ 1) are both independent and irreducible. 261. Types of degree four may all be expressed in terms of the independent forms {abY {acY {ady. If \ or /Lt be less than 2, then as in the previous paragraph the index of (ab) or (oc), as the case may be, can be increased at the expense of the index of (ad). 328 THE ALGEBRA OF INVARIANTS [CH. XV Thus, since the reducible covariant {aby {cdy-^ = {ahf [{ad) - (ac)}«'-^ + (^ 9 ) («&)' («c)^ (ad)''-^-' -..., the covariant {ahy (ac) {ady''~^~^ can be expressed in terms of reducible forms and of covariants (abf (acY {ady-''-^ (/^ > !)• When both \ and //, are greater than unity, say \ + /jb = M, we may express, by means of Stroh's series § 64, the products (aby (acY in terms of the following three sets : (i) {ab)^,(ab)^-'(ac),...{aby{ac)^-*; (ii) {ac)^,(ac)^-'(ab); (iii) (bc)^, (bc)^-' (ab). The products contained in (ii) and (iii) need not be considered, for the corresponding covariants can be expressed linearly in terms of reducible covariants and of covariants in which the number of factors involving a, b, c only is greater than \ + fi. These latter forms can be dealt with in the same way. Thus we see that ultimately we can express all the covariants of degree four in terms of reducible covariants and of such as have the factor (ab)*. Further we have seen that we may suppose the coefficient of (ac) to be greater than unity: hence all covariants of degree four can be expressed in terms of reducible covariants and of covariants of the form (aby {acY (ad)" where X-«j^4, fi-^2, v-^1, and the arrangement of the letters a, 6, c, d has been fixed beforehand. 262. The theorem can now be proved in general by induction. We shall assume that all covariant types, of a system of binary forms of infinite order, which are of degree n + 1 or less, can be 261-262] TYPES 329 expressed linearly in terms of reducible covariants and of covariants of the form (aoi)^' {aa^Y^ . . . (aa„)*'' , where \<^'^\ \2H:2""', ... Xn<i:l, and the arrangement of the letters a, a^, cto, ... a„ is fixed. For degree ?? + 2 we have only to consider the covariants of the form (aoj)^! (aa-.y^ . . . (aa,i+i/''+' = (aa^^y^ {aa^^-^ R. Now {aa^^R is of the same symbolical form as a covariant of degree w + 1 of the system ; hence, using the result for that degree, we may, if Xa < 2'*~^ express it in terms of covariants of the same form but for which the index of {aa^) is not less than 2"~\ and of reducible covariants. In the same way,- if Xj < 2'*~\ the index of {aa^ in the product (aoi)^' R can be increased. Let Xi + X, = J/ <{: 2 . 2"-S N = 2"-^ ; then, as before, by means of Stroh's series all products (aai)^ {aa.^ can be expressed in terms of the M-\-l following products : (i) {aa,yi, {aa,yi-^ {aa,), . . . {a(hf^{aa^) ^^-^, (ii) {aa,y^, (aa,yr-^ (aa,), . . . (aa.y^-^-'-' (aa,y-\ (iii) (a, aoy^, {a,a,y^- 1 (aa,), ...{a, a,y^-^+'- (aa^)^-\ The products contained in (ii) and (iii) need not be considered, for the corresponding covariants have factors of the [form {aa^R where p < 2"~^ ; hence these covariants can be expressed in terms of reducible forms and of products which contain a greater number of factors involving a, a^, a., only. The products contained in (i) all contain the factor {aa-^^. Hence all covariants of degree n + 2 are expressible in terms of reducible forms and of covariants which have the symbolical factor (aaiY^. But these can, by an application of the assumed result for degree n + 1, be expressed in terms of reducible forms and of the covariants (aa^Y^iaa^y^ ... (aan+i)^"+\ where Xi^2", \<i;2»-\ ... X^+iH:!. 880 THE ALGEBRA OF INVARIANTS [CH. XV The theorem is then true for degree n + 2 if it is true for degree n + 1; it has been proved for degrees three and four and is therefore true for all degrees. 263. It should be noticed that it has not been proved that the covariants i^etained are irreducible. It is practically certain that this is so, but no rigorous proof has yet been given. The number of covariants retained which are of degree n + 1 and weight iv may be found as follows. If i.e. if 'iv<2^^ — l, all the covariants are reducible. If w*i;2'^ — l, the covariants retained are of the form (aaiY" (aao)-"" . . . (acin) . R, where R is any product (aai)^' (aa,/^ . . . (acinY" , and Xi + X2 + ...+X„ = w-2" + l. Hence the number required is the coefficient of ii;'*'-2"+i in the expansion of (1— ic)~" — this being the number of homogeneous products of dimensions w — 2" + 1 of n letters. This is equal to the coefficient of a;*'' in the expansion of This generating function for perpetuant types is the same as that obtained by Macmahon's methods. 264. The results thus obtained for perpetuant types are of great use in obtaining either the types or the ordinary covariants of a binary form of finite order. All that was required in the course of the argument was that the weight of the covariant under consideration should not exceed the order of the quantic — or quantics. Thus any covariant of weight w and of degree h of the binary n-ic which is such that 2«-i -\>wi^n, is reducible. In § 114 the system of forms A^ for a single binary form of order ^12 was discussed. The above considerations of weight, 262-265] TYPES 331 alone shew that the following forms which were there retained are reducible {aby {hcf (cdf (de), (ahf (bcf {cdf {de\ {ahf (be) (cdy (deY ; the argument applies to the last of these three covariants only if the order of the binary quantic «a;" is greater than 12, it will be seen later that this is indeed reducible for the 12-ic. The remaining forms of the system would not be reducible as perpetuants and hence we cannot hope to reduce them for forms of finite order ; the two forms (aby (bey (cdy, (aby (be) (cdy, however, are congruent mod. (aby, as will be presently proved. 265. The theorem for covariants of forms of finite order corresponding to that which has been proved for perpetuant types is the following*. All covariants which are of the first degree in the coefficients of each of the quantics can be expressed linearly in terms of (i) covariants ofthefoi-m (flja.)^! (a.as)^- ... (as-ias)'^^, where \i ^ 2«--, \, ^ 2«-s, ... \s^l, and the arrangement of the letters a^, a.^, ... as is fixed ; (ii) covariants which have a symbolical factor (aj,akf(akai)'"''-^; (iii) products of covariants of lower total degree. The proof of this theorem follows that for perpetuants very closely. We first assume that it is true Avhen the total degree of the covariant considered is not greater than S — 1, and prove it when this total degree is S. Now the covariants to be considered can be expressed in terms of transvectants (a,;^,G,_,Y (I), • Young, Proc. Lond. Math. Soc. 1903. 332 THE ALGEBRA OF INVARIANTS [CH. XV where Cj-i is a covariant of the first degree in the coefficients of each of the quantics a^ **', a, ^^, ... as "« . On the assumption made (7s_i can be expressed linearly in terms of covariants of the second class, of covariants of the form and of products of covariants of lower total degree. If Cs-i is of the second class the transvectant (I), and each of its terms, must be of the second class. If Cj_i is a product of two covariants P, Q, then the total degree of P being less than S we may express it in terms of covariants of the form {ahY^{hcY^...{fgyi, and of covariants of the second and third classes. If /Lt + /ii be greater than the order of the form a the trans- vectant contains a term of the second class ; if it be not greater than this order the transvectant contains a reducible term. If P is of the second class the transvectant itself belongs to the second class ; and if P is of the third class, we may take one of its factors and proceed as before. If C5_i belongs to the first class, we may take C5_i = {a^(hY^ (asaiY' ■ • • (^s-i ^sY^ , then when Xs + M -^ ?^2, the transvectant contains a term belonging to the second class ; but when Xg + /"■ < n^ it contains the term {a^a^Y (aatts)^" {azCuY' ••• (a«-i«6)^«, and hence covariants of this form alone need be considered. 266. Let us now, for the sake of shortness, write Then we shall proceed to shew that, if /c < 2*"=', the trans- vectant {{a^a^Y ((^2a3Y, a^'^'-^^q/Yy^x can be linearly expressed in terms of covariants of the second and 1 265-267] TYPES 333 third classes, and of covariants which contain a greater number of factors involving aj, a^, a-i only. The two sets of covariants (aiCt-iY (a^a-iY, and {(a^a-iY, '^s/O''* where /* + /c has a fixed value and k takes all possible values less than 2^""-', are equivalent. Hence if (aia^Y = ax"'^"-""", the above transvectants may be linearly expressed in terms of the following, Any one of these transvectants is a covariant of unit degree in the coefficients of each of the 8—1 quantics it can therefore, by hypothesis, be expressed in terms of covariants of the form (aos)'^' {a^aiY' ■•• {a^-iCi^Y^-i, where fi, ^ 2«-^ /., ^ 2«-^ . . . fi,_, ^ 1, and of covariants belonging to the second and third classes. Thus the number of factors involving aj, a^, a^ only, can be increased when k < 2^~^ It should be noticed that covariants of the second class here include those which contain the factor (a„a)"'(aa;k)"'+"-^-2'^-''. It is easy to see that such a covariant belongs to the second class in the enunciation ; for, we may suppose v^ iii-'r n,, — 2/i — J/, ni ^ n.2, and therefore v'^iii — fx, the covariant considered then contains the factor 267. The covariant (a^a^Y («2«3)^' {c-iaiY" ••• {as-iasY^-i is a term of the transvectant ({a^a.Y (a^a^Ys aC^'q/Y'y^x, and hence differs from the whole transvectant or from anv one of 334 THE ALGEBRA OF INVARIANTS [CH. XV its terms by covariants in which the number of factors involving Oi, a2, fts only is greater than \^ + fi. By § 266, we see that we may suppose that neither X, nor fi is less than 2*~^ ; and hence that Xo + /* -"t 2^-\ The covariant can be linearly expressed in terms of the covariants (i) (a:aO^^+^ ia,a,Y^+>^-Ha,a,), ... (a,a,f-'{a,a,y^^'^-^'-\ (ii) (a,a3y^+^ (a,a,y^+'^-' (a,a,), ... (a,asY^+>^-^'-'+^ {a,a,f-'-\ (iii) (a,a3)^^+^ ia,a,Y^+'^--'{a,a,), ... (a,a,)^+>^-'^'-'+^asay-'-\ Transvectants of a covariant from one of the last two rows with aj^*~^*qy'' can be expressed in terms of covariants which contain a greater number of factors involving aj, ag, as only. Hence we may ultimately express all covariants in question linearly in terms of covariants having a factor (aitto)^', where Xj -^ 2*"^, and of covariants belonging to the second and third classes. Proceeding, as in § 266, with the covariants which have a factor (wia^)^' where Xi -^ 2*~-, we see that all covariants may be expressed linearly in terms of covariants of the form (Oiaa)^' (ttaaa)^ ... (as-ias^s, where X^ ^ 2^-\ \, ^ 2^-^ ... X5 ^ 1, and of covariants of the second and third classes. Thus the theorem is true when the total degree of the covariant is 8, provided that it is true when this total degree is less than B. 268. It remains to shew that this theorem is true when the total degree is three. The covariants to be considered are (aittj)*' (Oaaj)^ (asOi)'*'. Unless Xj + X2 + X3 is less than each of the numbers Wj, n.^, n^, this covariant belongs to the second class. For let Xi + X2 + X3 ^ ?l2, then by means of the identity (astta) = - (ctiaa) - (a2«3) 267-270] TYPES 335 the above covariaut can be expressed in terms of the forms and belongs to the second class. If \i + X2 + ^3 is less than each of n^, lu, 7I3 the argument used for perpetuants may be repeated here word for word. The theorem is then true for total degree three, and therefore for any total degree. 269. If all the quantics are of the same order n we obtain a theorem concerning covariant types of a simultaneous system of binary n-ics. In this case the covariants of the second class contain a factor of the form (aby {bcy-'^, and hence a factor of the form (aby (bcy-i' (cay, where '^ "^ 1 ' /3 "i 2?i — 3/a, (see § 68). Hence all covariant types of a system of binary n-ics can be expressed linearly in terms of (i) Covariants of the form (ttj a,)^! {aM^y^ . . . (ag_i asys , where \ ^ 2^--, \, <j: 2^-^ ... X^ ^ 1, and the order of the letters is fixed beforehand, (ii) Covariants tuhich have a factor ofthefonn (aby (bey -^ (cay, where '^^ ^> p-^2n — S\. (iii) Products of covariants of lower total degree. 270. The theorem just proved expresses all irreducible co- variants, of grade ^ ^ , in terms of a certain number of forms, which, there is good reason to believe, are irreducible when ?i is infinite. If this be so, these forms are certainly irreducible for finite values of n. 336 THE ALGEBRA OF INVARIANTS [CH. XV However, it does not follow that we cannot express them in terms of covariants of higher grade or else of covariants of the second class. In this connection we shall prove the following : all covariant types of the binary n-ic can be expressed linearly in terms of covariants of the form where \ ^ 2^2, 7^ > Xj, X3 > Xj, ... X^^^ > ^s-i, and of covariants belonging to the second and third classes. Using the previous theorem we see that covariants of the form G = (aitta)^' (aattay^ • • • (cis-iasYs-i alone need be considered. Let /*! :^ 2fi2, then (7 is a term of ((flhOa)^" (a^a^y^ {a^a,Y^ ... (aa_xafi)^a-ia,^«-«*a5^» -«*-«»... )%=<,, and differs from any other term by covariants which involve a greater number of factors containing aj, a^, as only. Then by Stroh's theorem, we may express (oia^Y' {a^f^aY^ i^ terms of covariants of the form {abY' {bc)\ where \i-^2\2, \i + X2 = /*i + At2j and a, b, c are the letters a^, a^, a-j in some order. Let (abY' = a^^-^s we have then to consider covariants of the form (aas)*^ (oaa^Y' • • • {as-iasY^-i. If X2 < /Its we may consider the transvectant {{oia;Y'{(ha,Y'> (a,aeY' .•• (as-.asys-i a./'-'^ni, "-'^^''^^Y^y-x- Then (aaa)^ (asaiY' can be expressed linearly in terms of (agai)^"*''*', and of members of the sets {aasY {asa,)^+''*-', {m,Y {aiasY'+ '''-'', where k > — ~-^ . 270-271] TYPES 337 We may proceed in exactly the same way at every step, and so prove the theorem. It will be noticed here that the order of the letters in the covariants is not fixed. 271. The Maximum Order of a covariant. Let us con- sider the covariant types of a system of quantics of which none of the orders exceeds n. By § 265, these types can all be expressed in terms of covariants of three kinds. Consider the covariants of the second kind. These contain a factor of the form We may suppose that X -^ n^ — X; the order of the covariant is then /ii + ^i, — 2X, :^ Wi < n. Now let us introduce a new symbol, for each covariant whose order does not exceed n. Covariants of the second kind are thus at once reduced in degree. Covariants thus reduced may them- selves be expressed in terms of covariants of the three different kinds. The covariants of the second kind may again be reduced, and so on. Hence finally we have expressed the system of covariants in terms of covariants of the form (aitts)^' (a^ttsY^ ... (as-ias^s-i, where \,^ 2^--, \^ '2^-\ ... Xs-i^l, — tti "', ttj "s ...as "« being either members of the original system of quantics or covariants of that system whose order does not exceed n — ; and of products of covariants of lower degree. The covariant of maximum order must then be of the form where X, < 2«-^ X, < 2«-^ ... Xs-i < 1, and ni, n^, ... ns are all equal to or less than n. The order of this for a given value of 8 is a maximum when Xl = 2«-^ \.3 = 2«-^ ... Xa_i=l, ni = ?i„ = . . . = Hs = n. G. & Y. 22 338 THE ALGEBRA OF INVARIANTS [CH. XV In this case the order is nS - 2 (1 + 2 + ... + 2«-2) = 7iB -2^ + 2. The maximum order is then the gi-eatest of the numbers n, 2n-2, Sn-6, ... w8 - 2* + 2, .... It is easy to see that if n = 2^ + ni, where ?ii<2^ then this maximum order is (\ + l)(2^ + w0-2^+i + 2 = (\-l)2^ + ?i,(\ + l) + 2. Comparison with perpetuants shews at once, that if the results for these are absolutely accurate, then the maximum order just obtained is always reached — even for a single quantic of order n except for the case n = S. Ex. (i). The covariant (ab)^ (be) (cd)* {def of the twelvic referred to in § 264 is of weight 13 and hence must be reducible to co variants of the second and third classes ; it is evidently reducible in the usual sense of the woi^d. Ex. (ii). Shew that the following covariants of the ten-ic can be expressed in terms of reducible covariants and of covariants of higher grade : {ahf{hcf{cd), {ahf{hcf{cdf, {ahf{hcf{cdY{de). References to papers by Jordan and Sylvester on the problem of § 271 and allied problems regarding weight and degree will be found in Meyer. The limits hitherto given are much too high for large values of n. CHAPTER XVI. GENERAL THEOREMS ON QUANTICS. 272. In this chapter certain results are obtained by an applica- tion of the theory, or rather the notation of the theory, of finite substitution groups. So little knowledge of this subject is required, that, for the sake of readers unacquainted with it, we shall start from the commencement, and prove the few well-known theorems required. In the first place a function of 7i variables is under consideration ; the function J \^2, ^1 J ^3, • • • ^n) is derived from this by the interchange of the two variables x^ and Xo. The operation by which the latter function is obtained from the former is called a substitution, it is usually denoted by the symbol (xiX.2). Thus we may write J (X^, Xi, Xg, ... Xji) ^ {XiX^}/ {^Xi, Xo, X^, ... Xyi). A more general example of a substitution is the operation by which the arrangement of variables Xi , Xo , • . . Xji is changed to 3/1' 2/2, •.• 2/n, the y's being the variables Xi, x.2,...Xn arranged in some order. This substitution is often written (Xi Xo . . . fl/jj N ^ 2/1 2/2. ..2/n/' *^"^ U 7"'T) f^^" ^2' ••• ««)=/(2/i' 2/2. ••• 2/n)- \</i 2/2 •• • yn/ 22 2 340 THE ALGEBRA OF INVARIANTS [CH. XVI Substitutions which represent merely the interchange of two variables are called transpositions ; thus the substitution {xiX^ introduced above is a transposition. The product of two substitutions. Let Si, s^ be any two sub- stitutions of the letters w^, x^, ... Xn, the meaning here attached to the product s-^s^ is that it is an operation which when applied to a function of x-y, x^, ... Xn is equivalent to the operation first of ■S2 on this function and then of Si on the resulting function.- (The usual convention is that the substitution on the left is the first to operate, but the above is more convenient for our present purpose.) Thus The effect of s^ is merely to produce a rearrangement of the variables, the effect of s^ on the resulting function is to produce a fresh rearrangement; thus the product of two substitutions is a substitution. It will be seen at once that substitutions obey the distributive law, for Si [SiS^\j \Xi, X^i ... Xn) = Si [SnSsJ \Xi, X^, ... X^)} = L51S2J S3 J yXi , Xo, ... X,,i). On the other hand substitutions are not in general commutative, for example : \X1X2) \XiXi) J \Xi , X2, i^s) '^ \^i^a)j \^s> ^2> '^i) ~y \^3> ^i> ^2)) ^^y \*'2> "^S) •^]/* Any substitution can be represented as a product of trans- positions. For any rearrangement of the letters Xy, Xr,,...Xn can be produced, first by an interchange of Xi and one other letter by which Xi takes its new position, next by an interchange of x.^ and another letter by which x^ is brought to its new position, and so on. It will be found that a substitution can be represented as a product of transpositions in a great number of ways, e.g. \X1X.2) ^ (^2'^3^ V'^l*^3/ V'^2'''37 5 272] GENERAL THEOREMS 341 but the numher of transpositions in a product which represents a given substitution is ahuciys even or always odd. For consider the function A = 1 1 — li \Xf Xg) J 1 r, s the effect of any transposition operating on A is merely to change its sign. Hence sA=+A according as 5 is a product of an even or odd number of sub- stitutions. Substitutions will be called even or odd according as the number of transpositions of which they are composed is even or odd. Consider any rearrangement of the letters x^, x^y.-Xn', let Xr be the letter which takes the place of x^ ; Xg that which takes the place of Xf] xt that which takes the place of Xg, and so on; we must sooner or later arrive at a stage when x^ is the letter which takes the place of x^^ the last of the series. The substitution which replaces x-i^ by a-v, ^V by Xg, Xg by Xt and so on, and finally x^^ by x^ is usually written {x^^XrXgXt . . . Xy) and is called a cycle. It is evident from the definition that \XiXyXgXi . . , Xn) ^ yXfXgXi , .. X'uXi). The rearrangement considered may be produced so far as the letters in the cycle are concerned by operating with {x^x^XgXt . . . Xu). Let Xa be one of the letters not contained in this cycle, then we may suppose that x^ in the new arrangement takes the place of x^ and proceed as before. Thus we see that the rearrangement may be produced by operating with a number of independent cycles, i.e. cycles such that no two contain a common letter. Hence any substitution is equal to a pr'oduct of a number of independent cycles. That operation which leaves any function operated on un- altered is called the identical substitution and is written 1. The product \x-^x^) . yXiX^) = yXiX^) leaves every function unaltered, hence yX-^X^)' ^^ 1. 342 THE ALGEBRA OF INVARIANTS [CH. XVI Consider the rearrangement of the letters Wi, x^,... Xn produced by any substitution s ; there is a perfectly definite substitution which will change this new arrangement back to the old arrange- ment. This is called the inverse substitution of s and is written s~^. In virtue of the definition of s~^ we see that 8 Sj \Xi , X^, ... Xfi) =J (Xi , X^, , .. X^)) and hence s~^s=l. Again it is to be observed that the result of operating on the new arrangement of the letters wdth ss~^ is to leave it unaltered, hence SS-' = 1. Consider the powers of any substitution, S, S", S^, ... ', they are all substitutions, and since the number of different sub- stitutions of n letters is finite — in fact nl, the number of possible arrangements of those letters — these powers cannot be all different. Hence for some values of h, k, In virtue of the associative law, we can write o2om __ gl+m Hence s^"^^ = s*+'. Further, whatever substitution <t may be, o- . 5^ = o-s* ; hence if o- = (s~^)^ we see that and hence 1 = ^~^. We may suppose that k > h, hence among the positive powers of s we must find the identical substitution. Let p be the smallest positive index for which s^ = l, then^ is called the order of the substitution. The substitution a-sa~^ is said to be conjugate to s. i 272-273] GENERAL THEOREMS 343 In o- let the letter which replaces *> be denoted by a;/; let arranged in some f-^'l Xo ... Xj{\ ~\yi y^-.-yJ' where y^, 2/2. •• .yn are the letters «i, x.2,.. • *X)'^i cil Att, order. Now <T ^/«i X., ...xn\^fy, 2/2 W x.^...Xn] Vy/ 2/2' ...2/J' and Hence _i f^i X^ ... Xn \ \Xi X2 ... X^i J cr5a"~^ = \yi y-2 "'yn\fX-^ X^...Xn\ fX^' y2"'ynJ\yi 2/2-"2/JUi u/2 • • • t*/-) /3/1 2/2 '..yn\(Xi X^' ...Xn\_ /aji' iCg' . \yi y-2 ■■■ynJ\yi y-2 '-ynl V2/1' 2/2' • 2 • • • ^tt \ '/...2/J 2/»i Therefore o-scr~^ is the substitution which would be obtained by operating on the expression for s with the substitution a. It must then be a product of cycles each having the same number of letters as the cycles of s ; and is obtained from s by permuting the letters. Such substitutions are called similar'. Every substitution similar to s is obtained from 5 by a suitable permutation of the letters, and is therefore of the form crso-~\ Now if s be any cycle {xiX^ ...xjc) then S~'^ = (Xk . . . x^x-^) as may easily be verified ; and hence s~^ is similar to s. But every substitution is a product of a number of independent cycles, the inverse substitution is then the product of the inverse cycles ; hence any substitution is similar to its inverse. 273. If m substitutions Sj, 50, ..- Sm are such that the product of any two of them is itself one of the m substitutions, these m substitutions are said to form a group. Thus as may be at once verified i, (^^1^2^ 5 1, {XiX2Xg), {X1XSX2) are groups. The number of substitutions included in a group is called the order, the number of letters affected is called the degree of the group. 344 THE ALGEBRA OF INVARIANTS [CH. XVI Thus the two groups written down are of order 2 degree 2, and of order 3 degree 3 respectively. Now the n letters x^, sc.,, ... x^ can be arranged in n\ ways, hence the total number of substitutions affecting n letters is n!. These substitutions obviously form a group, it is of degree n and of order n\. This group is called the symmetric group for the n It is useful to have a symbol by which to denote this group, the symmetric group for the letters x-^, x.^, ... Xn will be written \X1X2 • • • Xn\ . More particularly this symbol will be used to denote the sum of all the substitutions of the symmetric group. Again the product of two even substitutions is obviously an even substitution, hence the even substitutions which affect n letters form a group. This group is called the alternating group. Let Sl, S^y • • • Sjfi be the members of the alternating group, and the remaining substitutions which affect the letters Xi, X2, ... Xn. Then these latter substitutions are all odd ; and hence the product of any two of them is an even substitution. Now if ti, t^, tz be any three substitutions and then t~^ ti U = tr^ ti U , and hence ^2 = ^3- By hypothesis the substitutions Si, s^, ... Sm, (Ti, <t2, ... a-m' a,re all different, hence the substitutions are all different. But the former set include all the substitutions of the n letters, hence the latter must do so too. Hence the even substitutions form the alternating group. And therefore 273-274] GENERAL THEOREMS 345 The symbol {osiX^ ... a;„]' will be used to denote the sum of the even substitutions minus the sum of the odd substitutions of the letters Xif X2, ... Xn • This will be called the negative symmetric group, and on the other hand will be called the positive symmetric group of the n letters. For example {•^l'^2'^3/ ^^ 1 "t" (•^1^2'^3) "^ C'^l'^3^2) ~ {•^v'^2) ~ ('^2'^3) ~ V^S'^l)' 274. As an illustration of the notation just introduced we remark that the determinant hi 62 a„ bn may be written • [ah ... k}' aib.2 ... kn, the substitutions being supposed to affect the letters and not the suffixes. Or adopting a double suffix notation we may write n, 1 <■% 2 a, 1 a. "2,1 2, 2 n — [ttiCl^ ... (Inj (ii,iCl2,2 '•• Cin,n> Cf'n, 1 (f'n, 2 • • • ^n, ?i where the first suffix only appears in the substitutions and is alone affected by them. Or again aP bP cP : a9 6? c5 ={a&c}'a^6?c''. a*" b"" e As an example of the use of the positive symmetric group, referring to § 44, we observe that the rth polar of the form aa;'* = a, a, ... a,. XX X 346 THE ALGEBRA OF INVARIANTS [CH. XVI may be conveniently written 275. Consider any function F of the coefficients of certain linear binary forms a^, /3x, jx, which is homogeneous and linear in the coefficients of each form separately. We may write where the ^'s are functions of the coefficients of the linear forms 7-8, ^x, ■■• of the same character as F. Then {a/3}'i^=(a/3)[<^.3-</>3]; i.e. {a^y F is the product of (a/8) and a function which does not contain the coefficients of oCx or /3a;. Again we may write F=l,Clr^syt<l>r,s,t, {r,S,t= 1,2), but here the suffixes r, s, t can never be all different. Now [a^i\' ci,f^s^t= %■ as at A- yS, 0t It Is It hence {a/37}'i^=0. In the same way if Wj , a, , . . . be any ^-ary linear forms, where and F be any function homogeneous and linear in the coefficients of each, {fitiOa ... Op]' F = «1, ] '^l, 2 Os, 1 Ota, 2 a, , i>p V^, I '^i?, 1 *p, 2 • • • ^, p where the substitutions affect the first suffixes only, and -v/r is a function of the coefficients of ^p+\' ^p+i^' 274-275] GENERAL THEOREMS 347 The expression | aia.2 ... oip | will be used as an abbreviation for the determinant just written down. Again, ii r>p, {a,a, ...or,}'#=0; and ii r <p, it is easy to see that where Ax is one of the determinants of the matrix *1, 1 ■*!, 2 °h, 1 "2, : tto ^, 1 ^r, 2 • • • ^r, p It is unnecessary to suppose that a^-^i, a^_2, ••• Or.p are the coefficients of a linear jo-ary form a^ . The facts just established are true if F is homogeneous and linear in each of any m sets of quantities 02,1 tl, 2 '2, 2 H,p So 0^, 1 '■m,p there being p quantities in each set ; and a one-to-one corre- spondence between the members of any two sets. Thus in particular : If F is a function homogeneous and linear in the coefficients of each of m binary n-ics m being greater than n+1, then {oi,a, ... oin+.^'F=0 (i), {aitta ••• a„+il'i^=| aiOEj ... Un+i \ . F^ (ii), {a^cc^ ... an]'F= \ a^oL, ... a„yS | (iii), where the substitutions affect the first suffixes only of the coefficients ^p,q} I *i«2 ••• (X-n+i is the determinant of n+\ rows and columns formed by the coefficients of the ?i + 1 quantics concerned ; I ttiOa ... 0^^^ ! is the same determinant with quantities /3o. A. ••• ^n replacing O^n+i, o> Otji+i, 1> ••• O^n+i, «j 348 THE ALGEBRA OF INVARIANTS [CH. XVI these quantities being homogeneous and linear in the coefficients of the quantics represented by ctn+i, ci,i+2, ••• o^; (^nd where F^ is homogeneous and linear in the coefficients of the quantics represented The first two of the above results are sufficiently clear from what has already been said. As regards the last we observe that where Ax is one of the determinants of the matrix ^'l, «!, 1 0^2, ^,1 *2, n '"n, ^i, 1 <^,n That is Aa may be taken to be the minor of On+i, k in the determinant and hence {ajfls ... OL,^YF= ajOo ... ct,,^ |. 276. Let the quantics of the last paragraph be represented symbolically thus (Or, 0, Or, 1, • . . Or, n][«?i, a;^)" = [tt^;*'-*]", (r = 1, 2, . . . m). Then the determinant | aia2 ... ««+! I is an invariant, for it may be written symbolically n (a'^'^a^"^) (r, s = l, 2, ... m; r'^s). r, s Hence if F is an invariant, then Fy^ is also an invariant. Again we can shew that if F is an invariant, then (/3o, A. .../3«][^i, a^2f is a covariant. For let Ar be the minor of a^+i, r in the determinant I ftjaa ... ttn+i |. Then since F is an invariant is also an invariant. 275-277] GENERAL THEOREMS 349 Now we know that Sa„+i, ,.-4^ and 2a,i+i,r( \x-^^~^x^ are both invariantive, and hence that the quantities Aq, Ai, ... An and x^'\ i^\x^'''-'^x.2, ...(jx^^^ form two cogredient sets. Therefore, since 2^^ J.^. is an invariant, 2y8y ( ) x^^~^x^ must also be invariantive. In other words {^o,^,,'-'^n\x„X.,Y is a covariant. If F were a covariant, and contained the variables x^, x^, then /8o, /3i, ... ^n would also contain these quantities; in this case the form (/3o, A, ... ^n\yi,y^T is a covariant in two sets of variables. 277. As examples of the use of the results just established we may instance the fundamental identity for binary forms {abc]'{ah){cd) = 0; that for ternary forms [ahcd]' (abc) (def) = 0, or {abcdV (ahe) (cdf) = ; those for quaternary forms {abcde}' (abed) (efgh) = 0, [ahcde]' (abfg) (cdhi) (ejkl) = 0, and so on. Let / be an invariant linear in the coefficients of each of 7i + 1 binary n-ics ; then with the notation of § 276 we have [aiCXa . . . an+i}' / = X i ttitta . . . Cln+i | where \ is some constant, possibly zero. If n is odd we may take I = (a<^' a®)» (a<^' a<'")" • • • (a'"' a<"+i')'*- In this case, provided the n + 1 quantics are all linearly inde- pendent, 350 THE ALGEBRA OF INVARIANTS [CH. XVI is different from zero, for all terms of / in which two coefficients have the same suffixes are destroyed by the operator, and the rest are obtained from (- 1) 2 ( n - 1 ) a „_i a _ „+i \~2^/ ""^ '^'^''"^- by interchanging both the first suffixes of pairs of brackets in all possible ways keeping the order of the three first suffixes un- altered — so that the first suffixes of any one bracket are always of the form (2r — 1), (2r) and in this order — and by then inter- changing the first suffixes inside individual brackets, each such interchange being accompanied by a change of sign. But both these operations are effected by — since the first only requires even substitutions. Hence {ajOs ... a„+i}'/ = X {aiOg ... a,i+i}'oi oa2,na3,iO'4,9i-i ••• a «-!« , , n+i = ± \|aia2 ••• Oii+il where \ is not zero. Hence when n is odd lajOa ... a,i+i| is reducible. Again, if n = 4, it is easy to see that this invariant is irre- ducible. Let us suppose that it be reducible, then I aia2a3«4a5 1 = 2/2/3 where If is an invariant of degree r. Now /a must be of the form (a'^^'a'*')*, where r, s are two of the numbers 1, 2, 3, 4, 5 ; hence and therefore {aia2a3a4a5r | aia2*3«4«s I = 5!|aia2a3a4a5| = 0. This we know to be untrue in general, hence the hypothesis, that the invariant in question is reducible, is false. 277-279] GENERAL THEOREMS 351 278. Let s be any substitution of the symmetric group {aia.2...an}, then {aia.^ ... «.„} s = {ctiaa ••• ««}• For {aia.2 . . . an] s contains ?i ! terms, which are all dififerent (if a and a are different substitutions, then crs and a's are also different), and are all members of the positive symmetric group {aitts ... On]. Similarly s [aiUo . . . a„} = {aiCi^ . . . «,J. Now any purely formal relation between substitutions will still hold good if the sign of every transposition be changed ; hence S [aitto... ««}'= ± {(11(1.2 ... ttnY according as s is an even or odd substitution, in particular («! a^) {a^a-i . . . an]' = - {aitta . . . a„}'. Again if (ajaa ••• ««} be any positive symmetric group and {aitt^bs . . . bm]' be a negative symmetric group, the two groups having a pair of common letters, then (aitts . . . an] {tti a^bs . . . b^]' ^ = {aia.2 . . . (In] {(ha^) [- (a^a^) {a^a^ba ... bm]'] = - (aja, . . . an] {a^aobs . . . 6,«}' = 0. Similarly {aiaa^a . . . b^]' {a^a.^ . . . an] = 0. Thus if two symmetric groups, one positive the other negative, have a pair of common letters, their product is always zero. 279. The following purely formal theorem enables us to establish various results relating to invariants. Let the letters a^, a<., ... an be arranged in any manner in horizontal rows, so that each row has its first letter in the same vertical column, its second letter in a second vertical column, and so on ; and so that no row contains more letters than any row above it. 352 THE ALGEBRA OF INVARIANTS [CH. XVI Thus for four letters a^a^a^a^ the five possible kinds of arrangement of the tableau would be a. Then form the substitutional expression 8 = '2GiGo ... G^;iri'r2' ... r^;' such that Gi is the positive symmetric group of the letters of the first row, G2 that of the letters of the second row, and so on, Gh being that of the letters of the last row ; and that F/ is the negative symmetric group of the letters of the first column. Fa' that of the letters of the second column, and so on, F^;' being that of the letters of the last column (in case a row or column contains only one letter, it is understood that the positive or negative symmetric group of a single letter is unity). Let us suppose that in the tableau considered there are eti letters in the first row, ofa i^i the second, and so on ; where owing to the conditions laid down ai + a2+ ... + «/, = 71 J «! < "2 <f: tta ••• <t: ka, «! = ^^ let Ta„ 02, ...a ^6 the sum of the n ! expressions S obtained by permuting the letters in the tableau in all possible ways, the numbers ai, ag, ... a/i of letters in the various rows remaining fixed. Then ^-^a,, aj, ... «^ ^a,, aj, ... o» = 1 where the summation extends to all possible values of the numbers «!, a.2, ...ah which satisfy the conditions (I); and ^tt,.aj, ... a^ is a numerical coefficient which can be uniquely determined. For two letters we have For three letters we have 1 = ^-^ 3 + i-' 2,1 + Fir-' 1.1.1 as can easily be verified. 279-280 J GENERAL THEOREMS 853 280. Let the T's be arranged in the order defined by the con- vention that Ta „. a comes before Tg, b. b , when the first of the differences which does not vanish is positive. Now if S be one of the n I expressions of which T^^^ ^^^^ ^ is the sum, then Ta,, a., ... a^ is obtained from S by permuting the letters in all possible ways and taking the sum. Hence if when S is expanded as a sura of substitutions, any particular substitution s occurs in it, then in T^^^ „„, ... a^ the sum of all the substitutions of the symmetric group of the letters a^, Oa, ... «„ similar to s must occur. Hence defining t^^^ p^^ ... ^ to be the sum of all those substitutions which are formed of k cycles of orders 0i, ^^, ... ^8^ respectively, it follows that Ta„ a,, ... a, = 2X^„ ^,, ... ,3, ^/3„ P,, ... P, (H), where \p^^ p,^^ ... ,3 is a numerical coefficient. If cycles of order unity, which are equivalent to the identical substitution, be introduced, we may suppose that the suffixes of tpu /3„, ... ;8 satisfy the conditions The t's are now defined by numbers which obey exactly the same conditions as those which define the T's. The number of t's must then be equal to the number of T's, let us say equal to M. Now the equations (II) may be regarded as a system of linear equations expressing the ^'s in terms of the T's. Hence if these equations are all independent tp„ /3,, ... fi^ = S^a„ a.,, ... a„ -^ a„ a.. ... a,, and in particular ■■■ ~ h, 1, ... 1 ~ ^^a-i, a-i, ... a,j -'a,, a.^, ... a^ • If these equations are not all linearly independent, there must be a relation of the form S5T=0. G. & Y. 23 354 THE ALGEBRA OF INVARIANTS [CH. XVI In order to prove the impossibility of such a relation, it will be shewn that (i) ^ai, «„ ... a, ^^„ p,, ... /s,,, = when Ta^^ a,. ... a comes after T^s^, ^s^, ... ^s^,, the T's being arranged in the order defined above ; and that (ii) n.. <..... a, + 0. Let S = FN be one of the n ! expressions of which T^^^ „^^ ... „^ is the sum, where P denotes the product of the positive symmetric groups, and iV that of the negative symmetric groups. Similarly let S' = P'N' be one of the expressions of which T^^^ p^^ ... p^, is the sum. If one of the groups of P' contains a pair of letters contained in any one group of N then, § 278, NP' = 0. Consider the tableau by means of which S is formed, «! is the number of letters in the top row, it is also the number of columns, and consequently it is the number of the positive symmetric groups in P. Again a^ is the number of letters in the second row, and hence «! — a2 is the number of columns which contain one letter only. Similarly o^ — otg is the number of columns containing exactly two letters, and in general «» — Of+i is the number of columns containing exactly i letters. If ^i > «! there are more letters in the first row of the tableau for S' than there are columns in the tableau for ;S^. Hence one group at least of the product P' contains a pair of letters belonging to the same group of N ; and therefore KP' = 0. In order that NP' may be other than zero, we must then have (Si ::^ «!• If this condition be satisfied but we see that there are more letters in the first two rows of the tableau for S' than can be arranged in the tableau for S with the condition that no three occur in the same column. In this case some group of N must contain three of the letters of the 280] GENERAL THEOREMS 355 first two groups of P', and therefore two of the letters belonging to one of these groups. Hence if then NP' = 0. Again if ^i + ^ii + ^3><^i + a, + ^3, some one group of N must contain four of the letters belonging to the first three groups of P', and again NP' vanishes. Proceeding thus we see that NP' is always zero unless We deduce that if the first of the differences which is other than zero is negative, then NP' is zero, i.e. if Ta„ a.,, ... a, comes after T^,, ^^, ... ^^, then NP' = 0', and hence T... ,. ... „, T,,. ,,, ... ,^ = (SPiY) (2P'N') = 0. Next we must prove that T T =T- =t For this purpose it is only necessary to shew that the coefficient of the identical substitution is other than zero. Now but every substitution is similar to its reciprocal substitution ; hence if s be any substitution contained in ^^„/3j, ... js ,s~^ must also be contained in this expression. It follows that in Ta^, a^, ... a^ both s and s~^ have the same numerical coefficient. In T^. „ . the only products which produce the identical substitution are those obtained when a substitution s is taken in the first T, and s~'^ in the second. Hence the required coefficient of the identical substitution is of the form IX-, which, being the sum of a number of positive terms, cannot be zero. Let us now suppose that there is a relation of the form 23—2 356 THE ALGEBRA OF IXVARIANTS [CH. XVI and let \T„ „ „ be the first term of this relation, the terms being arranged in the order defined above. Then [2£r]r„..„^;...„, = o. And hence by the relation (i) ^^\, a.,, ... a,, = ; therefore by (ii) X = 0. Thus no T can be the first term in such a relation ; in other words the equations (II) are linearly independent. Hence l = S^„,..,,...a,^a.......a, (HI). Q.E.D. 281. The coefficients in this series have been calculated* but as their values are not of importance for our present purpose we merely quote the formula • n (a,, — ttg — r + sy / n {a,.- as- r + s)\' U,, a,. ... a, = yuior + h-ry.) The substitutions SiSz and So^i are similar, for S1S2 ^ ^2 \^2^1/ ^2> their coefficients in the expansion of T^^^ „..„ ... a,, are therefore the same. Hence since T = 1PN it follows that also If r^^^pj ... ^^, = ^P'N' comes before Ta„ «, ... a^ it has been shewn that NP' = 0. Hence also P'i\^=0, and N'FNP = 0, and \%N'F][1NP] = {), and therefore ^/Si. iSj. ... ^v -^tti. o-i, - «A — ^- That is, the product of any two different T's is zero. * Young, Proc. Lond. Math. Soc. vol. xxxiv, p. 361. 280-282] GENERAL THEOREMS 357 Now multiply the relation (III) by T^^, a^, ... a,^, \ye obtain at once T —A 7*2 282. By polarizing the form once with respect to each of the sets of variables r,' (1) 7- (1) r (1) «i<2' a;2<2> ... a;,,,<2) (m) ^ (m) ^ (m) rp [in rf> tvo • • . ■^91 we obtain an expression which may be written = — {aWa<2» ... a<'"'}a'iVi'a'"a;<2) ... a"«>a;(m). If in F^ each of the sets of variables is replaced by the original set x^, x^, ... Xn, we obtain the form F from which we started. Neither the passage from F to F^ nor that from F^ to F affects the invariant properties of F, so that we may regard F and ^1 as equivalent. Similarly if /(a'^', a'^*, ... a""') be an invariant linear in the coefficients of each of m quantics of the same order, whose coefficients are ao^ ai'^ ... (r = l, 2, ...m), then [a^^^a^-^ ... a<'«>}/(a'i', a'^*, ... a<'">) is an invariant which may be obtained by means of Aronhold operators from an invariant of a single quantic. Again if P be the product of li positive symmetric groups, which between them contain all the letters a'^*, a'-*, ... a*'*", but no two contain the same letter, then P/(a"', a^-'\ ... a**")) is an invariant which may be obtained by means of Aronhold operators from an invariant of h quantics. 358 THE ALGEBRA OF INVARIANTS [CH. XVI 283. Peano's Theorem. Let F{a^^\ a^-\ ... a'"") be a covariant linear in the coefficients of each of m binary 7i-ics (ao"^, a,**-*, ... aj^'^ccu x^f; r= 1, 2, ... m. Operate on F with the two sides of the identity § 280 (lii.) 1 _ VJ T ' J. — ^-a.^.^, a.2, ... ttj -^ o., o.,, ... a^) then i^=S^.„^^,...„^T,,.^^,._^i^. Now T„„^,....^ = 2Pi^, where N has a factor of the form {a<«a(--=' ... a<^']'. If A > 11 + 1, then by § 275, [a^'^a^'^'' ... a(*'}'i'=0, if A = ri + 1, {a^a® ... ff,(A>}'2?^=| aWa<-' ... a<*) \.R, if A = 71, {a<"a® ... a<'^'}'i?'=| a»'a<=« ... a^^^Q \, where {Qq, Qi, ... Qn^Xi,x.^"^ is a covariant of the forms considered. Hence when h>n-\-\, Ta„a„...aF=0; when h = n + l, Ta^,a.„ ...a^F is a sum of terms each of which contains a factor of the form | a'^'a'^' ...a'"+^' j; when h = n, Tai, a^, ... a^^ 18 a sum of terms of the form j a^^^a^-^ ... a'"'Q |. Now the number of positive symmetric groups in P is h; hence by § 282, Ta^,a.„...a^F is a sum of terms each of which is obtainable by means of Aronhold operators from a covariant of only h different 7i-ics, Let us suppose that P is a covariant type which does not give any irreducible covariant, unless we are considering a system of more than n w-ics. Then if h<n-\- 1, Ta„ 02, ... a.^F is reducible for it is obtained by Aronhold operators from reducible forms : if /< > n + 1 ^ai.cHj, ... o*P= 0; 283-284] GENERAL THEOREMS 359 and if k = n + l, Ta,,a,,...a^F is a sum of terms each of which contains a factor of the form | a'''^''a^^'' ... a*""^^' |, and is thus reducible unless m = w 4- 1. Hence evei'y type of the binary n-ic which does not furnish an irreducible covariant for a system of n n-ics is reducible, with the possible exception of the invariant type I a^^^a^^^ ... a"^+i» |. Further it has been shewn that, if h = n, Ta^, a.^, ... ai^F is equal to a sum of terms of the form i a^^^a^^^ ...a^^^Q . But if the invariant | a^^^a^-^ ... a"'"*"'* | is reducible, the invariant is reducible in the same way. Hence when the above invariant is reducible every irreducible type furnishes an irreducible co- variant for a system of n — 1 w-ics. It has been shewn § 277 that this invariant is reducible when n is odd ; thus the covariants of any number of cubics can be obtained by means of Aronhold operators from those for two cubics. Similar results may be obtained in exactly the same way for ternary forms, or for forms involving any number of variables. Thus all covariants of any number of ternary ?i-ics may be obtained by means of Aronhold operators from the system for n + 2 , — 1 ternary vi-ics with the possible exception of the ()i + 2\ j rows and columns, each row being formed by the coefficients of one n-ic. Ex. Shew that the determinant formed by the coefficients of six conies is an irreducible invariant of the system. 284. Peano's theorem was first proved l)y means of an ex- pansion due to Capelli*, which is virtually the same as the expansion used here, but is expressed in terms of polar operators. * " Sur les Operations dans la Theorie des formes Algebriques," Math. Ann. Bd. 37, See also "Lezioni sulla Teoria delle Forme Algebriche," Ch. I. § xxiii. by the same writer. 360 THE ALGEBRA OF INVARIANTS [CH. XVI It is sufficient, in order to make the comparison clear, to point out that if / be a function linear in each of n sets of ^-ary variables, a^, a^, ... an', then where A^ is one of the determinants of the matrix ^,1 '^i.a ^,9 Ctg, 1 C^S, 2 '^a «'2,5 and in fact is equal to Ha„a^,...a^f where Dx being the determinant of the matrix d 9ai,2 '" d '" d(li,q d d Bag, 2 d '" da2,q d d d dahi caji dah,c obtained by writing ^ — for ar,s in each element of A\. (Jdr, 8 In the case of binary forms, if a^^h^^ be polarized so as to obtain a function of m + n sets of binary variables, and then the identity § 280 (lll.) be applied, we obtain a proof of Gordan's series, § 52, which is a particular case of the series of Capelli. The coefficients in Gordan's series are not apparent, but it is possible to obtain them by these methods*. Thus if /= aa;(i)aa;'2* . • • <*«'">> &j/'i' ^2/*2) . . • hy(n) , then T.„a,,...aj=0, when h> 2. If h = 2, we need only consider those expressions PN which are of the form PJV= {a;Wa;<2> ...a;""'y'^* •••2/'*'l {y**+'*?/''+^' •••2/'"*} {x^^^y^^'^^'Y ... {aj"'-^'^/"*'}'; * See Young, Proc. Lond. Math. Soc. vol. xxxiii. 284-285] GENERAL THEOREMS 361 and ill this case PJSf.f=P.(x^'^y^'+'^) ... (x^"-^h/''^)(ahy'-'a^(n-i+i) ... aa;(m)6y(i) ... byU) , which may be obtained by polarization from (xyy-^ {oh)"--^ a^^^-'^ 6/ In just the same way we see that, if x, y, z be ternary variables, ax"^by^CzP can be expanded in a series each term of which may be obtained by polarization from a term of the form X {xyzf (abcY (abxyY^ (bcxyy^ (caxyy^ a^^ bx^Cx^, where {abxy) = Let then tti Oj x^yz x^y^ a^ O2 x^y-i — Xiy^ (h bs x^y^-x^^yj III ^=^23/3 ^33/2 ^2 = ^3 2/1 ^1^/3 W3 = ^1^/2 ~ ^22/1 ) u, = is the condition that the point z lies on the straight line joining the points x and y. Hence Uj, a.^, u^ are really the coordinates of a straight line. We see then that all covariants of ternary forms which contain any number of variables can be expressed in terms of polars of covariants which contain the variables x and u only. The m's were introduced for geometrical reasons § 207, but it is now apparent that they are necessary to make the analytical theory complete, 285. Let a;'^', a;*^*, ... a;*'*' represent n sets of q-&vy variables ^i**"', ^•o*''*, . . . ^9*'"' ; r = 1, 2, ... /I. Then if i'' be a function linear in each of these sets, we see, in the same way as before, that PNF is a function obtainable by polarization from a function, not necessarily linear, of 1st the sing-le set 7' (1) 2nd the determinants of the matrix X.2 Xq^^ 362 THE ALGEBRA OF INVARIANTS 3rd the determinants of the matrix [CH. XVI '3) g,<a, ^ is) and 80 on, and lastly of the determinant «. (2) J- (1) tlyq .w '5' a^5*«' These may be all regarded as auxiliary variables. It will be useful to denote the variables of the 2nd set, viz. the determinants of the matrix r (1) by the letter 2«, w'ith appropriate suffixes. Similarly the third set will be denoted by gX and so on. Then in taking the complete system of concomitants of 5-ary forms we have q — l different kinds of variables which may appear. The geometric meaning of these variables is easy to obtain. A space of g- — 1 dimensions being under consideration, the variables cc or iX represent point coordinates : the variables ^ are line coordinates; the variables 3X are plane coordinates, and so on. The linear substitutions by which these auxiliary variables are transformed, when any linear transformation of the point coor- dinates is made, are easy to find. The variables {oc and q_iX are contragredient, and in fact •i* {X q — {X is an absolute concomitant. As a particular case, when q is even *, this leads to a relation between the variables qX. This remark 2 has already been illustrated for quaternary forms § 220. The case of binary forms is of course an exception. 285-286] GENERAL THEOREMS 863 286. Let ax^^\ «a;'-*, ... a^^"'^ be any 7i linear q-ary forms; and F a function linear in the coefficients of each. Then apply the formula F=2A T F where substitutions interchange the sets of coefficients aW, a<2), ... a'«'. Consider the form PNF, it is a sum of terms each of which is a product of determinants of matrices of the form ttj (1) a. (11 ...a^^' a, a. (2) a^ (2) __ ...a,<^'' (t) a-. CM ._ ...agC-" Moreover if iV" has two negative symmetric groups of degree i and Ai^x, Si,fi represent determinants from each of the correspond- ing matrices (the particular determinant being defined by the second suffix), then every term of NF has a factor of the form Hence every term of PNF has a factor of the form Ai^K-Bift + Ai^fiJji^\ as is evident when the tableau from which PN is constructed is considered. But this is a coefficient of the expression [tAiix][l.Biix] which is a covariant since the as are contragredient to the x's. Now if PN is a term of 2"^^. „.„... a,., N" contains a^ groups of degree h, Uh-i — cch groups of degree h — 1, and so on ; hence PNF is a linear function of the coefficients of concomitants of the form Thus every function of the coefficients of certain linear forms can be expressed in terms of coefficients of concomitants of those forms. It is unnecessary to assume that the function is linear in the coefficients of each of the forms, in order that the above theorem may be true. For the function can be made linear by means of 364 THE ALGEBRA OF INVARIANTS [CH, XVI Aronhold operators, and after the above process the original coefficients can be restored without affecting the invariantive properties in question. Further the forms considered may be symbolical, and we at once deduce that every integral function homogeneous in the coefficients of each of certain q-ary forms can he expressed as a linear fuTiction of the coefficients of the comomitants of those forms. APPENDIX I. NOTE ON THE SYMBOLICAL NOTATION. As we have said in § 82 the notation used in this work is really equivalent to Cayley's hyperdeterminants. The great advance made by the German school lies in the possibility of transforming symbolical expressions, and, of course, in the proof that every invariant form can be represented as a combination of hyperdeterminants. The reader may feel the need of justifying directly the results obtained by manipulating umbral expressions and accordingly we shall indicate how the whole theory can be made to rest on differential operators. There are different ways of doing this. Salmon has remarked that, /being a binary form of order n, since we may regard/ as being equal to where "^"'''W.) [fyj ' Hence we may suppose that 5 d and that the final operation is on A nl' 366 THE ALGEBRA OF INVARIANTS [aPP. I Any symbolical expression can be thus at once transformed into one exactly like it but involving only diflferential operators, e.g. if then (a^y aaT-' ^x'''' is ( _s^ ?Lyf^ 1+^ ir~7^ 1+^,1^^ fe \dy1dZ2 dzjdyj V ^dyi ^dyz) \ ^dz-t, 'dzj m\n\ (see § 82). In any calculation we may omit the operand while we trans- form the operator. After this the reader who wishes to do so will have no difficulty in developing the theory of the symbols when they are regarded as differential operators. For another method see Kempe, Proc. L.M.S. vol. xxiv. p. 102 and Elliott, Proc. L.M.S. vol. xxxiii. p. 231. Some interesting general remarks on the underlying principles of the symbolical notation will be found in Study, Methoden Terndre Formen ; and some very curious remarks in Lie-Scheffers, Vorlesungen Uber Gontinuier lichen Grilppen, p. 720. This is the most convenient place to give a brief explanation of the so-called Chemico- Algebraic theory — an idea originally due to Sylvester which has perhaps attracted more attention than its intrinsic merits deserve. In this theory an atom in chemistry corresponds to a binary form in algebra, and the valency of the atom to the order of the form. To each unit in the valency of an atom, in the chemical theory, a bond is supposed to correspond, and each such bond can connect the atom in question with an atom of valency one such as Hydrogen. Thus Oxygen is of valency two, and there exists a compound OH2 which is written graphically H— 0-H, there being two bonds proceeding from O and one from each H. Since each unity in the order of a form gives rise to one possibility of trans vection with another form, the analogy is evident APP. l] NOTE ON THE SYMBOLICAL NOTATION 367 — if we have a binary quadratic o^^ and two linear forms Aa;, ^x tbe formula OHg corresponds to the algebraic expression iph) {oh') an invariant of the three forms. Then Carbon being of valency four we have the compound (Marsh Gas) H I CH4 or H— C— H I H and this corresponds to the invariant {ch){ch'){ch"){ch"') of a binary quartic and four linear forms. The four hydrogen atoms in CH4 are supposed on chemical grounds to occupy similar positions* in the structure of the compound and hence CH4 is more naturally like {chy an invariant of a quartic and a single linear form hx. Then the compounds CH3CI, CHoClo etc. may be supposed like the invariants (chyick), (chy{cky where k is written for CI and we see in the chemistry an analogue to polarizing in algebra. Guided by the above the reader will have no difficulty in writing down an invariant corresponding to any graphical formula however complicated — in fact the algebraic form of the invariant is only a different (perhaps a more concise) way of writing down the chemical formula. Difficulties arise when we recollect that some atoms have apparently different valencies illustrated by S in SO., and H2S, for of course a binary form can have only one order. Gordan and Alexeleff suppose that the corresponding algebraic form is then polarized. Thus S in SOo would correspond to Sx'^ and S in HoS to Sx^Sy- and now the degree available for transvection is two. * For an explanation of this and the other chemical facts we have referred to see Scott, Chemical Theory, chap. vi. 368 THE ALGEBRA OF INVARIANTS [aPP. I The idea has been developed by various writers in the direction of making the algebraic methods graphical (references in Meyer) and lately Gordan and Alexeleff * have written several papers in which the algebra is applied to chemistry ; the papers were criticised by Study f and from the objections and the replies the reader may be able to form his own opinion. We venture only to remark that the wonderful feature of the algebra is the capacity for reduction, and that, unless there is something cor- responding in chemistry, the whole theory seems to be no more than a superficial analogy. It is of course certain that a reducible invariant often corresponds to a stable compound — moreover the general features which lead an algebraist to suppose a form reducible and a chemist to suppose a compound unstable are as nearly opposite in character as they can be. It has been stated in Chapter i. § 21 that there are two ways of obtaining relations between concomitants symbolically expressed, viz. : (i) By means of the fundamental identities {he) ax + {ca) b^ + (ab) Cx = 0, (6c) {ad) + {ca) {bd) + {ab) {cd) = ; (ii) By means of the fact that a concomitant is left unaltered when a pair of letters which refer to the same quantic is inter- changed. We give here a demonstration of the fact that all relations may be thus obtained :|:. Let F{Gi) = be any identical relation (supposed rational integral and homogeneous) between concomitants Ci of any system of binary forms. Let F{Ci) = 'ZPj each term Pj being itself a concomitant. Also let (^0,^1, ... ^nfc.^2)" = a^" = 6=r"=-.. be one of the quantics of the system. The introduction of the symbolical notation may be effected by operators like "■"ax/ ''■""■''=ai. + ••• + "•" ax * Wiedemann's Annalen der Physik, 1899, 1900. t In Wiedemann. X Cf. Gordan, Invariantentheorie, Bd. ii. § 117. APR l] NOTE ON THE SYMBOLICAL NOTATION 369 operating on F{Ci). These operators do not destroy the property that F{Ci) considered as a function of the coefficients of the quautics is identically zero. Hence if F{Ci) =XPj becomes "StUk, where each term Hk is a symbolical product of factors of the forms (ab), Ux, considered as a function of the symbolical letters. Owing to the manner in which the symbolical letters were introduced no distinction was possible between two letters which refer to the same quantic, hence XHk is unaltered by any inter- change of two such letters, and therefore the second method of obtaining relations between forms will have no effect here. Let Oi, «2, Us, ... Ur be the symbolical letters, and n^.n^, ... rir the orders of the quantics to which they refer — if the forms Ilife are covariants we shall suppose that a,-,2 = *'i, ctr,i = — ^2 ^-nd that 71,. is the order of 11^. Then applying the theorem of §47, x being replaced by «i and 1/ by ctj, we obtain d where SIIj;*-" does not contain da- Now SUk = 0, hence every term of this series is identically zero. For if the jth term be the first which does not vanish, we may divide by {aicuy, and then put aa = di, whence \;[Sni<^'] = o. But this series is merely obtained by repeated use of the fundamental identities (see § 46), hence the reductions used so far belong entirely to the two classes mentioned. It may happen that '^TLk^^^ is not identically zero considered as a function of symbolical factors. In this case we may\^ipply the same process again. Each time we do this the number of letters in the function under consideration is reduced. Hence in (r — 2) steps at most the expressions are reduced to expressions which are identically zero when considered as functions of the symbolical factors — for when only two letters are left there is only one possible symbolical factor. Thus the identity F{Ci) = is made to depend entirely on the two fundamental methods of reduction. G. & Y. 24 ^ "' • r 2,11^ = "S^Xjioiasy a^ i=o L APPENDIX 11. ON WRONSKI'S THEOREM AND THE APPLICATION OF TRANSVECTANTS TO DIFFERENTIAL EQUATIONS. The form of Wronski's theorem used in § 189 is not the usual one, but the determinant, which there vanishes, can be transformed in the following manner. If f^aj\ then 5-^^^ — = 7 ^ aa;"-^-'^ a^^a.f, and __ I (w — A, — /i-) ! It follows that where the ^^'s are numbers depending on n, r, X and fju. Now in § 189 the determinant whose typical row is dx^"-' dxi'-'dx^'"' dx^"-' vanishes, hence so also does that whose typical row is dx/' '"'dxr^dx,''"''' dx/' APP. Il] WRONSKl'S THEOREM S71 On using the value found above for the typical row of the determinant becomes and since this determinant vanishes we infer, on modification of the columns, that the determinant whose typical row is dec/' dx/-^'"'-^ vanishes. If we replace x.^ by unity in the /'s, this is the usual form of Wron ski's determinant. For the sake of completeness, we shall give an easy proof of Wronski's theorem. If Ml, M2. •■• Un+i be 71+ 1 functions of a single variable so, and the determinant whose rth row is **'■' 'd^' dc^""dc^ vanish, then there is an identical relation of the form XiMi + \Mo + . . . + X„+i lln+i = 0, where the X,'s are constants. In fact the vanishing of the determinant is the condition that the w's should be solutions of the same linear differential equation of order n, say We have therefore to prove that such an equation cannot have {n + 1) linearly independent integrals. The theorem is easy to establish when n is unity, so we assume it true for n — 1 and proceed inductively. Now «i being a solution of the equation of order n, write y = Ui fivdx. 24—2 372 THE ALGEBRA OF INVARIANTS [aPP. II It is quickly seen that w is given by an equation of order n — 1, say and this equation is satisfied by d^ fu^ d I'uA d^ fUn+A dx \Va) ' doc \uj ' '" dx\ Ui J ' hence by hypothesis there is a relation of the type Integrating and multiplying up by u^ we obtain a relation of the form between the (n+ 1) functions u, so Wronski's theorem is completely established. The application of the result to the (r + 1) functions / shews that if x^ be replaced by unity, there is a relation of the type \fi + ^2/2 + • • • + Xr+ijr+i = 0, and then making each / homogeneous again by the introduction of flJa, the result quoted in § 180 follows at once. The device of changing from differential coefficients with respect to two variables to differentiation with respect to a single variable is often useful. For example, the rth transvectant of is Vr = (a6)'-a^'"-'-6a,'*-*'. Thus x^^yfr = {a-Jb^^ — aaftiiCsy^*"'"'"^*""'" ^ {n-r) \ dj ^ {n-r + \) \ (m-1)! d'-'f 80 ~ n ! dx{ ^ n\ ml dx/~'^ dxi m ! ox/ APP. Il] WRONSKI'S THEOREM 373 Applying this to the nth transvectant of two forms / and (f), each of order w, we can reduce the problem of finding a form / apolar to <f> to the solution of a linear differential equation of order n ; it follows at once that there are not more than n linearly independent forms, and moreover from the algebraic theory we infer that all integrals of the equation are polynomials in Xi. Conversely a differential equation whose coefficients are poly- nomials can be reduced to a relation between transvectants. To give a simple example, consider the equation daf- dx the P's being polynomials in x. If ^1, 02 > <\>i be three forms of orders r^, rj, r^ respectively, and / be a form of order n, ' v/^p^r - ,, (^ _ 1) <P3 g^^2 ^,,^ 5^^ 9^^ + ^^ (^r, -ly dx,^ • Now replacing x.2 by unity as usual, we can choose <^i, ^2,^3, so that is the same as for the transvectant relation is 1 . ^f _l_dl #3 ^ 1 .rf^0, n (w — 1) ''^^ cifa?!^ m-g cla;i c?a?i ^3 (rg — 1) •' c^a?!^ and we must have \ , df 1 ^d<f>. ., „ 03 = -Po, rj (w — 1) 1 2 rf03 _ p ?2 ^ nrg dxi 1 6^02 1 (^'03 ^ p 374 THE ALGEBRA OF INVARIANTS [APP. II The first equation gives <Jjs and its order is that of Pq, the next equation gives (f>2 and then the last gives ^i ; hence the trans- formation is always possible, but of course the coefficients depend on the order of the form / that is chosen to represent y. References to further developments in connection with Differ- ential Equations will be found in Meyer's Berichte and in Klein's lithographed lectures on Linear Differential Equations of the Second Order. APPENDIX III. JORDAN'S LEMMA. In Chapter iv. gi-eat use was made of this theorem: — If x + y + z = 0, then any product of powers of x, y, z of order n can he expressed linearly in terms of such products as contain one exponent equal to or gi-eater than -^. In § C4 a proof due to Stroh is given. We shall now give a simpler proof iu which a much more general theorem is incidentally established. The general theorem may be stated as follows : — It ax, Ox, Cx, ... be a system of r distinct linear forms and a, /3, 7,... be r positive integers satisfying the relation a + + y + ... = n-r + l, then it is impossible to find binary forms A, B, C, ... of orders a, /3, 7, . . . respectively such that ax"-' A + hx"-^B + Cx''-yG + ... = (I). In fact, suppose that such an identical relation exists and operate a + 1 times with a d J. 376 THE ALGEBRA OF INVARIANTS [APP. Ill It is easily seen, by making a.^ zero, that i)"+^ annihilates a binary form of order n only when that form contains the factor tta;"""; hence since a^, bx, Ca;, ... are all different, we have l^n-p-a-i B' + c^'^-y-a-i C" + . . . = (II), where B', G',... are of orders /3, 7, . . . respectively and do not vanish identically unless B, C, ... do also. The relation (II) is of the same form as (I) except that r is changed into r — 1 and n is changed into ?i — a - 1 for + y+ ... = n-r-a + l=(n-a-l)-{r-l)+l. Now a relation of the type (I) is impossible when r = 1 for any value of n unless the form A (the only one occurring) vanishes identically, hence by induction it is impossible for all values of 71 and the theorem is established. The forms A, B, C,... involve a + 1, y8 + l, 7+I, ... arbitrary coefficients respectively, or in all a + ^ + y+ ... + r = {n + l). Any binary form can be expressed linearly in terms of ()i 4- 1) linearly independent forms of the same order, and since there is no identical relation of the form aa,"-« A + 6a;"~^ B + Cx''-y C + . . . = 0, it follows that any binary form of order n can be expressed uniquely in the foi-m a^"-«^ + 6it"~^ B + Ca^-y + ... , where a^, bx, Cx, ■-• are all different. A, B, G,... are of orders a, y9, <y, ... respectively, and a + y8 + 74-...=w-r+l. Consider now three linear forms ^, y, z, where z = — {x-\-'y) and a;, y are the variables. APP. Ill] Jordan's lemma 377 It follows from the above that if a + i3 + y = n-2 then any homogeneous expression of x, y, z of order n can be expressed in the form a;"-" P + ^"-^ Q + ^»-y R, where P, Q, U are of order a, /3, 7 respectively. In other words, changing a into n — X, into ?2 — /i and 7 into 71 — V, any homogeneous product of order n of cc, y, z can be expressed in the form .'/^ P + 2/'" Q + z" R, where X + fx, + v = 2n + 2, and the expression is unique. This is Stroh's generalized form of Jordan's lemma given in § 64, and the lemma itself follows at once since we can always choose integers X, fx, v satisfying the relation \ + /A + i/ = 2n + 2 , 2n 2n 2n and ^^3' ^^"3' ^^"3* On expressing P in terms of x and y, Q in terms of y and z, and R in terms of z and a?, it follows that any homogeneous product of order n of x, y, z (a; + 2/ + 2 = 0) can be expressed linearly in terms of a?", x'^~^y, x'^~'-y-, ... x^y^^~^ yn^ yn-iz, y'''-"-Z-, ... y^^z"'-*^ 2", 0''-i a^, 2»-- a;^ ... z" x''-" and the same would still be true if z were changed into x in the second row. The reader will have no difficulty in modifying tlie above so as to obtain another proof of the fact that the general system of apolar forms constructed in § 178 contains n linearly independent forms. APPENDIX IV. FURTHER RESULTS ON CO VARIANT TYPES. The expression given in § 262 for perpetuant types, may be used to determine the perpetuants when the forms are supposed not all diflferent. The result (using the notation of the paragraph referred to) for one quantic is ^-8-1 = 1 + ^S-i \s-2= 2 + ^5-1 + ^5-2 X, = 2^-^ + 2(|5-: + rfi-2+...+^2+^0> where all the ^'s are zero or positive integers. For two quantics the covariant can be written in the form ((ha^)^^ (a^asY^ . . . (tti_ia,-)"»-i (aAYiib^b^Y^ (hobsf^ . . . (bj^.bjfj-i, where the a's refer to one quantic and the b's to the other ; the indices satisfy the conditions 2a, = 2i+i-^ + 2 (^,_i + ^,_+...+ 10. For the proof of these results see Grace, " On Perpetuants/ Froc. London Math. Soc, vol. xxxv., p. 219. APP. IV] TYPES 379 The reasoning of this paper for the case of a single quantic may be applied to types, with the result that all perpetuant types of degree B may be expressed in terms of products of types and of types of the form (ai a.f^ (aoUs)'^^ . . . {as-i asYs - 1 , where the X's satisfy the conditions laid down above for types for a single quantic, except that and where the order of the letters ctj, a^, ... is no longer fixed. The method of §§ 262, 265 can be extended to the case of any concomitant which is linear in the coefficients of each of certain binary forms Oi '^1, a, '^s ... 0,5 '^5. Thus any such concomitant can be expressed by repeated transvectiou in terms of (i) forms of the type (( ... {{{a^a.^^m^YmiY^... a^-O^s -2 05)^5-1, (ii) reducible forms. The relations satisfied by the indices are however somewhat complicated. They are given as follows : — A reduction (in the sense of § 265) is always possible when \i < 2*~^ unless one of the following conditions is satisfied : \ + S (2X; + 2^-^- - n,-+0 > n, - (2«-'- - 1 ), X, + S (2Xj + -I'-J - nj+,) > n, - (2«-^ - 1 ), where B> i >j > I, but i and j are otherwise unrestricted, as also is the number of terms under the sign of summation — in particular there may be none. In general, the condition that Xj = 2*~- — a may not mean a reduction is that certain positive integers, /i(0),/.(0),/;(0); /i(|i),/.(liX/3(^i). ^1=1,2; /i(|i, ^2\M^i, ^.),/s(^i, ^.), |i=l. 2: f, = l, 2; M^^, I., - Ir), /.(I:, I2, ... U /adi, I2. ... ^), ^,= 1, 2: |, = 1, 2:...^r=l, % 380 THE ALGEBRA OF INVARIANTS [aPP. IV can be found such that a =/ (0) +/.(0) +/3(0) +/(1) +/(2) m) =/i (r) +/. (^) +/3 (?) +/(!, 1) +/(?, 2) /(?!, f., ... L) = A(|., f., ... ^)+/.(|:, |„ ... I.) +/3(|l, ?2, .... |r)+/(fl, I., ... Ir, l)+/(|a, |„ ... ^r, 2); (where the quantities /(fi, ?2) ... fr) are positive integers which satisfy /(?:. h, ... ^, l.+O > 2^-^- -/,(?„ ?„ ... ?.)-/3(?.. |„ ... ^) when ?i + ?2 + . . . + |r + f r+i + r is odd, and /(?!, ^.. ... I.. ir+x)>2«^- -/,(!„ r., ... ^•)-/3(ri. r.> -. ^) when ?i + ?2 + • . • + |r + ?r+i + »" is even) ; V+2 + 2«-'-^ + </>l(?i, ?2, ... rr)>«i+/i(?n ?2, ...?r)-l*, V+2 + 2«— ^ + «^2(|l, ?2, ... |r)>«2+/2(?x, I2, ... ?r)-l, X,+, + 2«-'-^-/(?„ |„ ... ?.)>rj,+3+/3(|i, I2, ... ?r)-l ; where, finally, the ^'s are defined by the laws <A,(ii. r.. - r.) +/(?:. ?., .- ?r)-</>jfi, ?2, ... r.-i) is zero if ?/ + ?i + ?„ + • • • + l^r is even, but if this sum is odd, the expression considered = (2V+i + 2«—-7l,^,)-2{y(f , ?„ ... ?.-:)-/(?>. I2. ... ir% and <^,(0) = -a = -/(0), <^, (1) +/(!) = /.,- 2 {/(O) -/(I)}, (^2(1)+/(1) = 0, <^,(2)+/(2) = 0, 02 (2) +/(2) = /^. - 2 {/(O) -/(2)1. * In case /i(fi, lai ••• fr) is zero, this inequality need not be satisfied; this remark applies also to the other two inequalities. INDEX. Alexeleff 367, 368 Algebraically complete system for binary forms 150 for ternary forms 265 Anharmonic ratio (see Cross-ratio) Apolarity of binary forms defined 213 the a linearly independent forms of order n apolar to a binary n-ic 214 there are s linearly independent forms of order n apolar to ii-s + l given binary ;/-ics 220 the cubic apolar to three given cubics 220 forms of different orders 225 the 71 linearly independent forms of order m apolar to a binary in-ic (tKni) 226 the forms of order n apolar to a given 7K-ic {n<m) 228 application to canonical forms 229 Apolarity of ternary forms of two conies 299 of cm'ves in general 303 Curves which possess an apolar conic 304 Cubics which possess an apolar conic 312 Quartics which possess an apolar conic 313 Conies apolar to fundamental conic 308, 310 Argand diagram, Interpretation of invariant properties of binary forms by means of the 208 Aronhold 70, 84, 357 Aronhold operator is invariantive 31, 32 Associated forms 148, 157 Ex (v) Ausdelmungslehre 279 Baker 298 Bezout 318 Brianchon 280 Canonical forms 233, 234, 235 for quintic 231 for sextic 231, 232 Capelli 359, 360 Catalecticant of binary forms of even order 232 of ternary qaartic 313 Cayley 3, 84, 131, 365 Chemico-algebraic theory 366, 367, 368 Ciamberlini 274 Circle, director 302 self-conjugate 302 Class 255 Clebsch 31, 132, 138, 147, 150, 157, 255, 265, 271, 274, 326 Clifford 288 Cogredient transformation defined 36 sets of variables, forms with 40, 60, 61 Combinants 314-318 Complete systems 85, 109 relatively 109, 110 completeness of svstem derived from two 111, 113, 114 Complex variable 208-212 Concomitant, Mixed 255 Conic Binary quadratic and geometry of 235-240 Ternary quadratics and geometry of two 288-294 Polar 251, 256 Apolar, see Apolarity of ternary forms Conjugate lines 300, 302 points 300 Contragredience defined 36 Fundamental property of 37 of point and line coordinates in ternary forms 255 Contravariant 255, 256 Convolution defined 40, 41 Coordinates, Line 255, 361 Copied forms 80, 81, 82 Covariants of Binary forms defined 8 Symbolical expression of 16, 17, 18 with several sets of variables 40 Covariants of ternary forms. Symbolical expression of 250 382 INDEX Covariants of degree three 62-71 of degree four 71-74, 76, 77 Cross ratio defined 186 cross ratios of roots of a quantic and the invariants 189-191 of the roots of two quadratics 192, 193 of the roots of a quartic 204-206 Cubic, Binary Irreducible system for one 89-91, 122 Irreducible system for quadratic and 164, 165 Irreducible system of types of 323-325 Syzygy for 96 Transvectants of 96, 97, 194 Geometry of 193, 194 Pencil of cubics K/"+\H 194-197 Geometry on Argand diagram of 211, 212 Eange formed by three points and one of their Hessian points is equian- harmonic 196, 211 Associated forms for 157 Pencil of cubics having a given Ja- cobian 241, 242 Cubic apolar to three given cubics 220, 221 Rational plane cubic curve represented by 221-224 Twisted cubic curve and binary cubics 240-242 Cubic, Ternary has three linearly independent apolar conies 304 Canonical form for 312, 313 Cyclically projective range 196 Darboux 306 Degree defined for binary forms 9 for ternary forms 255 Determinant formed by coefficients of 7J + 1 binary n-ics is a reducible invariant if n is odd 350 is an irreducible invariant if n = 4 350 Differential equation whose coefficients are polynomials can be reduced to a relation between transvectants 373 Diophantine equations 102-106 Discriminant of binary forms 191 of binary quadratic 86 of binary cubic 194 of binary quartic 199 Tangential equation of curve found from the 266, 267 Double points 223, 224 Duality, Principle of 256, 257, 267 Elliott 61, 150, 167, 366 Equianharmonic range formed by cubic and one of its Hessian points 196, 211 Faltung 40 Forsyth 264, 265 Franklin 131 Fundamental theorem for binary forms 25-27, 31-35 for ternary forms 258-264 Genei-ating function 330 Gordan 61, 67, 68, 85, 128, 131, 132, 138, 150, 169, 274, 294, 326, 360, 367, 368 Gordan's series Existence proved 45, 46 Coefficients of 53-55 Inversion of 59, 60 Deductions from 55-57 Gordan's proof of Hilbert's lemma 178- 182 Gordan's theorem 101, 118-121 Grace 326, 378 Grade defined 62 of covariants of degree three 66, 67 of covariants of degree four 77 Classification by 119-121 Grassmann 279 Groups, Continuous 6, 7, 38 Finite substitution. Introduction to 339-344 Positive symmetric group 345, 346, 357 Negative symmetric group 345-351 Gundelfinger 132, 232 Harmonic quadrilateral 301 range 191, 192, 203, 207, 272 Harmonically eoncyclic points 209 circumscribed conic 300 inscribed conic 300 Hesse 306 Hessian defined 3 is a covariant 3, 16, 17 identically vanishes when quantic is a perfect Jith power and conversely 193, 235 points of a cubic 195, 196, 207, 211, 212 of binary cubic defines the double point on the corresponding cubic curve 224 of ternarv forms 249 Hilbert 145, 169 et seq., 274 nomographic ranges 187, 188 transfoimation 253 Hyperdeterminants 83, 84, 365 Identities, Fundamental for binary forms 19, 321, 368 INDEX 383 Identities, Fundamental (cont.) for ternary forms 257, 258 in general 349 Inflexion, Points of 222-224 Invariants defined 7, 8 of binary forms, Symbolical expres- sion for 14, 15, 16 of ternary forms, Symbolical expres- sion for 250, 251 Inversion 208 Involution 203, 238 Irreducible system of forms 85 solutions of Diophantine equations 102-105 Jacobian of binary forms 8, 77 is a covariant 8, 9 of a Jacobian and another form is in general reducible 78 Product of two 20, 78-80, 140 vanishing of 193 of two quadratics is harmonic to both 192 of pencil of cubics 241, 242 of two quartics apolar to a third 245 Any combinant of two binary quad- ratics is an invariant of their Jacobian 315 of ternary forms 249, 251 Jordan 62, 126, 321, 375, 377 Kempe 366 Klein 374 Lie 38, 366 Line coordinates 255 Linear forms Invariants of a binary system of 33-35, 85, 86 Invariants of any finite binary system and a 160, 161 Invariants of a ternary system of 275 Geometry of 279 with any number of variables 346 MacMahon 326, 330 Maximum order 337 Meyer 245, 317, 368, 374 Mixed concomitants 255 Number of independent co variants of degree three and given weight 65, 67 Operator fi for binary forms : Effect on symbolical product 23-25 Transvectauts defined by 46 and generalised transvectauts 83 and hyperdeterminants 84 Expansion of fi'" { {^v)"P} 173-175 Operator for ternary forms : Effect on symbolical products 259-261 and transvectauts 295, 296 Order 9, 255 Osculating plane 188, 243 Parametric representation 236, 305 Pascal 240, 280 Peano 821, 358, 359 Pencil of rays 188, 189 of cubics {Kf+ \t) 194-197 of cubics (Xi/i-f Xa/^ 241, 242 of quartics 197-208* Perpetuants 326-330, 338, 378 Polar conic 251, 256 Polar forms Definition and symbolical expression of 12, 13, 345, 357 are covariants 39, 40 Adjacent terms of 44 Difference between two terms of 42- 55 Expansion of one term in 55 Poncelet's porism 245 Porism 245, 288 Projection 253 Quadratics, Binary possess invariants 1, 2 Complete system for one 86, 121 Complete system for any number 161, 162 Syzygies for any number 162-164 Covariant Types of 321-323 Associated forms for one 157 Geometry on straight line for 191-193 Geometry on Argand diagram for 209-211 and geometry on a conic 235-240 Complete system for cubic and one 164, 165 Complete system for quartic and one 168 Any finite system and one 165-168 Combinants of 315 Quadratics, Ternary Complete system for one 275-277, 298 Complete system for two 280-286 Geometry of two 288-294 (see also Apolarity) Quadratic, Quaternary Complete system for 270 Geometry of 271-273 Quadrics 272 Quantics defined 3 Quartic, Binary Complete system for 91-94, 122 Transvectauts for 97-100, 197 Syzygy for 98, 200 Discriminant of 198, 199 Quadratic factors of sextic covariant of 200-203 384 INDEX Qnartic, Binary {cont.) Associated forms for 157 Geometry and pencil of 197-208 Cross ratio of roots of 204-207 defines a pencil of cubics 241, 242 defines a twisted quartic curve 242- 245 Irreducible invariant of five quartics 350 Quartic, Ternary condition for an apolar conic, 304 Canonical form for 313 Quaternary forms 269-273, 349, 362 Quintic, Binary Elementary transvectants of 94, 95 Complete system for 128-132 Linear covariants of 133 Invariants of 133, 136 Values of transvectants of 135, 137- 140 Syzygies for 143, 144 Typical representation of 145-147 Associated forms for 148-150 Canonical form of 229-231 Eeciprocation 291 Rectangular hyperbola 303 Eepresentation, Typical 145-147 Reye 245 Richmond 245 Rosanes 245 Salmon 365 Schlesinger 307 Scott 367 Self-conjugate circle 302 triangle 238, 288 Seminvariant 28-30 Sextic, Binary Complete system for 150-156 Canonical form of 231, 232 Associated system for 157 Straight line, Geometry on 183-189 Stroh 71, 140, 142, 144, 152, 164, 326 Stroh's series obtained 62-64 deductions from 65-71 scheme used for 68 used 99, 100, 328, 329, 336 Study 366,368 Stlife 62 Stunn 245 Superosculation, Points of 244 Sylvester 131,366 Syzygies for Quadratics 162-164 for Cubic 96, 194 for Quartic 98, 198 for Quintic 143, 144 Stroh's method of obtaining 140-143 Reducibility of 144 of second and higher kinds 145 Finiteness of system of 177, 178 Tangential equation 301 Transformations, Linear defined 4 form a group 5, 6 of binary symbolical pi'oduct 13, 14 Geometrical interpretation of 184-189, 208, 251-254 of ternary symbolical product 247, 248 Modulus or determinant of 1, 2, 14-17, 21-23, 248, 249 Translation principle for ternary forms 265-268 for quaternary forms 271-273 Transvectants defined 46 Calculation by means of polar s of 47-49 Adjacent terms of 49 Difference between two terms of 49-52 System obtained by transvection from two finite systems 107-109, 111 Generalized 83 of ternary forms 296, 297 Triad 194 Types, Covariant of quadratic 162, 321-323 of cubic 323-325 of binary n-ics 335, 336, 379 Perpetuant 326-330 Tvpical representation of binary quintic 145-147 Ueberschiebung 46 Unicursal curve 188 Variables, Different species of 361, 362 Von Gall 326 \Webb 186 Weber 173 ;®eight 15, 17, 28 Wronski 233, 370-372 Young 331,356,360 CAMBBIDOE : PRINTED BY J. & C. 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